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Minimal-Area Metrics on the Swiss Cross and Punctured Torus

The closed string theory minimal-area problem asks for the conformal metric of least area on a Riemann surface with the condition that all non-contractible closed curves have length greater than or equal to an arbitrary constant that can be set to one. Through every point in such a metric there is a geodesic that saturates the length condition, and the saturating geodesics in a given homotopy class form a band. The extremal metric is unknown when bands of geodesics cross, as it happens for surfaces of non-zero genus. We use recently proposed convex programs to numerically find the minimal-area metric on the square torus with a square boundary, for various sizes of the boundary. For large enough boundary the problem is equivalent to the “Swiss cross” challenge posed by Strebel. We find that the metric is positively curved in the two-band region and flat in the single-band regions. For small boundary the metric develops a third band of geodesics wrapping around it, and has both regions of positive and of negative curvature. This surface can be completed to provide the minimal-area metric on a once-punctured torus, representing a closed-string tadpole diagram.


Introduction and Summary
A class of minimal-area metrics on Riemann surfaces is the key ingredient in the definition of a field theory of closed strings [1]. In this paper we continue the investigation we began in [2], where we used the tools of convex optimization [3] to formulate programs whose solutions are these minimal-area metrics. Related ideas from convex optimization have also proven useful in other contexts [4,5]. The goal of our work in [2] was to address the minimal-area problem [6,7] that asks for the conformal metric of least area on a Riemann surface under the condition that all noncontractible closed curves be longer than or equal to a fixed length s that can be chosen arbitrarily. The length s is the systole of the surface, the length of the shortest non-contractible closed curve. All curves of length s are geodesics and are called systolic geodesics. In general the surfaces have marked points and curves cannot be moved across those points. The minimal-area metrics are known for spheres with arbitrary number of marked points [8,9]. They are also known in some regions of every moduli space of Riemann surfaces with genus greater than or equal to one and some number of marked points.
The extremal metric is expected to have systolic geodesics that cover the full surface. Near the marked points the systolic geodesics are circles on a flat semi-infinite cylinder, with the marked point at infinite distance. Each point on that cylinder belongs to just one systolic geodesic. On the rest of the surface there can exist regions where points belong to just one systolic geodesic, and regions where points belong to several systolic geodesic. In general the surface is covered by multiple bands of intersecting and non-intersecting systolic geodesics. It has been shown that the conformal metric in a region covered by a single band of systolic geodesics is flat [10,11]. Generically, however, the metric is not known as soon as systolic bands intersect. This will happen for surfaces over finite subsets of each moduli space M g,n of Riemann surfaces of genus g ≥ 1 and n marked points. 1 All known minimal area metrics are locally flat, have curvature singularities, and arise as the norm of Jenkins-Strebel quadratic differentials [12].
A variant of this minimal area problem, as well as a particular case of Gromov's isosystolic problems [13], asks for the Riemannian metric of minimal area on a twodimensional surface of fixed topology and fixed systole. Here one minimizes over the conformal structure. Calabi [14] proved that for this problem there are no regions covered by a single band of systolic geodesics. Moreover, in regions covered by exactly two bands of systolic geodesics, the metric is flat and the geodesics in these bands are orthogonal to each other. Calabi described the foliations using calibrations, and this approach was elaborated upon and extended by Bryant [15]. The proofs, however, do not apply in the conformal case. In fact, extremal conformal metrics have regions covered by single bands of geodesics and the conformal metric in regions covered by two bands of geodesics is not necessarily flat, as we will see. For more information on systolic geometry see [16].
For this paper we wanted to identify the simplest possible surface for which we have intersecting bands of systolic geodesics and the extremal metric is not known. A good candidate is provided by Strebel's challenge: Find the extremal metric on a Swiss cross with two length conditions [17]. A Swiss cross, defined as the region of the complex z plane shown in Fig. 1, is a planar figure that can be built bringing together five unit squares or by arranging two identical rectangles perpendicular to each other. A Swiss cross has four edges: a left edge (x = 0, y ∈ [1,2]), a right edge (x = 3, y ∈ [1, 2]) a bottom edge (x ∈ [1,2], y = 0) and a top edge (x ∈ [1,2], y = 3). We constrain the lengths of all curves beginning on the left edge and ending on the right edge, like γ 1 . Similarly, we constrain the lengths of all curves beginning on the bottom edge and ending on the top edge, like γ 2 . We thus have constraints on curves that necessarily intersect. For the Swiss cross shown in the figure, whose arms have parameter length 3, a convenient choice of systole is s = 3.
Perhaps surprisingly, the extremal metric for the Swiss cross was unknown (some attempts to find it were made in [18,19]). Not only that, one was also lacking a qualitative understanding of the properties of the extremal metric. It was not known if it is locally flat (except at special points) or, if curved, whether the curvature is positive or negative, constant or non-constant. Moreover, while one must have two bands of saturating geodesics, it was not known if each band would cover the full cross or if the bands intersect over some proper subregion of the cross. We will provide answers to all these questions.
Recalling that a conformal metric ρ(x, y) is defined by the length formula ds = ρ|dz|, with z = x + iy, we note that the flat constant metric ρ = 1 is admissible because it satisfies all length conditions. Since it has area 5 we know that the Swiss cross minimal area A satisfies A < 5. On the other hand, it was shown in [18] that A > 9/2, so we have 4.5 < A < 5. (1.1) The Swiss cross is an excellent problem to try the new programs proposed in [2]. As formulated above, however, it is not a closed string theory minimal-area problem. We will show that the solution of the Swiss cross problem actually provides the minimalarea metric on a torus with a boundary. The torus is obtained by identifying the left and right edges and the top and bottom edges of the Swiss cross. The resulting surface (after cutting along the dotted lines in Fig. 1 and rearranging the parts) is shown in Fig. 2. We have a square torus with an aligned square boundary. In this torus, the minimal-area problem constrains the curves homologous to γ 1 and the curves homologous to γ 2 to be longer than or equal to s = 3.
A torus with a boundary belongs to a moduli space relevant to open-closed string theory. In fact, by making the arms of the Swiss cross of arbitrary length (while keeping their width equal to one) we can introduce a modulus to the Swiss cross. That modulus is also a modulus for the torus with a boundary, as it controls the ratio of the size of the boundary to the size of the torus. The metrics we will find would be relevant to some We constrain the curves homologous to γ 1 and those homologous to γ 2 to be longer than or equal to s = 3 vacuum graphs in open-closed string field theory. 2 Equipped with a modulus, chosen as the size of the square boundary, we can also study the moduli space of tori for the case of small boundary, where we require that curves homotopic to the boundary are longer than or equal to s . In the limit when the size of the boundary goes to zero we find the metric on a once-punctured square torus. This metric is relevant to closed string theory-it belongs to the one-loop once-punctured string vertex V 1,1 of the closed string field theory -and was previously unknown. A discussion of this vertex can be found in section 2.1 of [2].
Our work begins by applying the two programs in [2] to find the extremal metric and the value of the minimum area for the Swiss cross/torus-with-a-boundary problem. As explained in detail in [2], while the closed string minimal area problem constrains homotopy classes of curves, the programs are most simply formulated to constrain homology classes of curves. For the present problem, however, the distinction has no effect; the two homotopy classes of curves we wish to constrain belong to two homology classes, and constraints in these homologies do not impose additional length conditions on curves. Using the max flow-min cut theorem the length conditions on non-contractible curves are implemented by a set of calibrations (closed one-forms u with fixed periods and satisfying |u| ≤ 1). This turns nonlocal length constraints on the metric into local constraints, making the problem tractable for numerical analysis. This first program, called the primal, minimizes the area while allowing the calibrations that constrain the metric to vary. Introducing Lagrange multipliers for the constraints and eliminating the original variables, we obtained a dual program. In this program we maximize a functional defined in terms of a set of functions on the surface with prescribed discontinuities across curves that represent the homologies that are constrained. By strong duality the minimum of the primal and the maximum of the dual must agree. Both programs are implemented by defining a lattice on the Swiss cross and writing a Mathematica program for each case, using the functions FindMinimum and FindMaximum to obtain the extremum. Our results indicate that the minimum area A is A = 4.675 145, (1.2) with an error of ± 1 in the last digit. Both the extremal metric and the systolic geodesics are determined by the programs. In Fig. 3 we show the systolic geodesics on the s = 3 Swiss cross and its torus version, respectively.
Here are a few features of the extremal metric, in terms of the Swiss-cross presentation: 1. The saturating geodesics form bands; a horizontal band (blue) and a vertical band (red). The bands intersect over a central region. The boundaries of these bands are piecewise geodesics going through the corner points of the central region. 2. The surface has four regions covered by just one band of systolic geodesics. The metric is flat in these regions and the geodesics reach the edges orthogonally. 3. The metric has positive non-constant Gaussian curvature throughout the two-band central region. The integral of Gaussian curvature over this region is equal to 2π , the same as that of a round hemisphere. 4. The boundary of the central region is the locus of negative line-curvature. On each arm of the Swiss cross this curvature integrates to −π , for a total of −4π . 5. The metric diverges at each of the four corner points of the central region. This divergence straightens out the boundary, making its extrinsic curvature at those points zero.
Our results suggest it may be challenging to find an explicit closed-form expression for the extremal metric. Nevertheless, the metric has a number of remarkable properties. Both the bulk curvature and the line curvature integrate to simple quantities. Moreover, the value of the extremal area satisfies a simple relation. The optimum of the dual program includes parameters ν 1 and ν 2 , defined as the values of the discontinuity of the functions associated to the two constrained homologies. Because of symmetry, in the Swiss cross the two parameters are the same, both equal to some value ν. As demonstrated in [2], section 7.3, the value of ν at the optimum is the height of the band of systolic geodesics measured on a region where it is the only systolic band. In the Swiss cross this height is the length (on the extremal metric) of any of the edges. The dual program predicts that the minimum area A is 2ν s , as if the surface was built from two rectangles, one for each band, each of systole length and height ν! Somehow the complicated metric arranges matters for this to hold. The positive curvature on the central region is reminiscent of Pu's result [22] for the constant positive curvature minimal-area metric on RP 2 .
We explored the moduli space of the square torus with a boundary. To do so it is convenient to describe the torus as a square with an edge of size 1 and a square boundary with an edge of size h. The systole is then chosen to be equal to 1 for all values of h. In Fig. 3, for example, the edge of the boundary has size 2, while the edge of the square has size 3. With an overall scaling we conclude that h = 2/3 for this torus. The full moduli space of the square torus with a boundary is 0 < h < 1. As h → 1 the boundary is as large as possible, and this corresponds in string theory to long-time open string propagation. For all h > 2/3 the minimal area metric is qualitatively similar to the h = 2/3 metric whose properties were enumerated above. As we consider h < 2/3 no qualitative changes occur until h = h (1) 0.4201. At this point the arms of the Swiss cross have become short enough that the central region hits itself (see Fig. 27). For h < h (1) we find a surprise: in the region with two systolic bands, the extremal metric displays both positive curvature and negative curvature.
As h → 0 the boundary turns into a puncture, or marked point, and we have a oncepunctured square torus. This is the simplest surface in closed string theory for which the minimal area metric is unknown. We study this torus by considering the torus with a boundary for small values of h. In closed string theory all non-contractible closed curves must be longer than or equal to s . This includes the closed curves that go around the boundary. In the previous discussion of the surfaces we only constrained the curves in Fig. 4. Systolic geodesics on the torus with a hole removed. The hole is a square with side length 1/16 that of the torus. There are geodesics in three different homotopy classes: 1-geodesics (blue), which run horizontally; 2-geodesics (red), which run vertically; and 3-geodesics (black) which wrap the hole two homologies. But even then, the curves going around the boundary are in fact longer than or equal to 1 as long as h ≥ h (2) 0.24469. For those values of h, therefore, the extremal metric solves the minimal area problem that also constrains curves homotopic to the boundary. For h < h (2) , however, the minimal area metric with two constrained homologies fails to satisfy the length condition for curves homotopic to the boundary. This condition must now be imposed explicitly. The closed curves going around the boundary, however, are homologically trivial. Indeed, any such simple curve γ divides the surface into two regions, one that includes the boundary and the rest R; the curve γ is the boundary of R, thus trivial in homology. As explained in [2] to impose constraints on non-contractible curves that are trivial in homology, we need to pass to a suitable cover of the surface in which the curves in question are no longer trivial. For the square torus a four-fold cover is particularly convenient, as it allows us to preserve some of the symmetries that simplify the problem. We can then write both primal and dual programs on the covering surface and obtain minimal area metrics for the original surface.
As soon as h < h (2) the minimal area metric displays three systolic bands: the first two along the original two homology classes, and the third wrapping around the puncture. In Fig. 4 we show the systolic geodesics for a torus with boundary size h = 1/16. Here are the main features of the metric: 1. There are three bands of saturating geodesics: a horizontal blue band, a vertical red band, and a black band surrounding the boundary. The black band does not intersect any other band. The blue and red vertical bands intersect each other in a simply connected region, whose boundary is made of piecewise geodesics. 2. The black band represents a flat, finite-height cylinder of circumference 1. One of the boundaries of the cylinder is the boundary of the surface. The other boundary is the last geodesic in this homotopy class. In the limit h → 0, this last curve defines the local coordinate at the resulting puncture. This is the information required for the calculation of off-shell string amplitudes or the evaluation of the string field theory master action. 3. The black band is surrounded by four triangle-shaped flat regions covered by single bands of geodesics (red and blue, alternating as we go around the black band). We encounter no curvature along the last black geodesic, although the lines of curvature on the boundary of the two-band region, meet it tangentially at four points. 4. Over the two-band region the metric exhibits both positive and negative Gaussian curvature. The latter appears near the midpoints of the edges of the figure. The integral of Gaussian curvature over the two band region is equal to 2π , the same as that of a round hemisphere. 5. The boundary of the two-band region has eight segments, four red and four blue, each carrying an integrated line curvature of −π/2. Thus we get −4π from line curvature. There are no point curvature singularities 3 and therefore the total integrated curvature is −2π , consistent with an Euler number of −1. 6. The metric is finite and non-vanishing everywhere on the surface. This confirms the absence of point curvature. One can also confirm that all curves in other homotopy classes are longer than or equal to 1, making this a full solution of the closed string theory minimal area problem.
The methods of the present paper can also be applied to the study of conformal metrics of non-positive curvature on Riemann surfaces. These metrics are also of interest for string field theory and exhibit surprising features revealed by the numerical study: not all of the Riemann surface is necessarily covered by saturating geodesics [23]. 4 This paper is organized as follows. In Sect. 2 we review the results of [2] that are needed for most of this paper. The review is brief and for more details the reader should go back to the original reference. In Sect. 3 we discuss the minimal area problem on the Swiss cross with its length conditions on open curves and the minimal area problem on the associated torus with constraints on two homology classes of closed curves. We show that the solutions of these two problems are the same. In Sect. 4 we discuss the primal and dual programs on the Swiss cross, explaining in detail the role of symmetry in simplifying the problem and reducing it to a fundamental domain one-fourth of the size of the surface. We explore the use of other conformal frames to study the extremal metric, in particular a frame in which the fundamental domain is a pentagon and the central corner on the cross is straightened out. Our results for the tori with constraints on two homology classes are presented in Sect. 5. We give the extremal area, metric, and curvature of the solution for h = 1/2. We also discuss the moduli space of these surfaces, as determined by the size h of the boundary when the systole is assumed to be 1. The important case of the once-punctured torus is studied in Sect. 6. We show in detail the construction of the relevant covering spaces, which are also tori with boundaries, and exhibit the corresponding flows (dual to the calibrations). We build the requisite doubly periodic calibrations in terms of the Weierstrass zeta function. We include in this section the results for the extremal metrics. We conclude with an "Appendix" where we give details on the numerical discretization and discuss the convergence of our results.

Review: Convex Programs for the Minimal-Area Problem
In this section we review and summarize the results of our earlier paper [2] that are relevant to our present discussion. In that paper we considered a minimal-area problem with constraints on homotopy classes of curves and replaced it by one with constraints on homology classes, showing how the former could be solved by an extension of the latter. The minimal-area problems were formulated as convex programs.
Let M be a Riemann surface equipped with a fiducial conformal metric g 0 μν . The general conformal metric on M takes the form g μν = g 0 μν , with a positive function on M. The norm (of either a vector or a one-form) with respect to g 0 μν is denoted by | · | 0 , and with respect to g μν by | · |. We also define the fiducial volume form ω 0 = d 2 x √ g 0 and the volume form (2.1) The homology version of this problem considers non-trivial homology 1-cycles C α ∈ H 1 (M, Z), with α ∈ J an index labeling the cycles. We aim to minimize the area of M subject to the constraint that all representatives of each cycle C α have length at least α . In analogy to the homotopy MAP above, we write In both of the above programs the length condition is applied to an infinite set of curves.
In the first reformulation, we replaced the length constraints-conditions for the metric that are nonlocal on the surface-by local constraints that define, for each C α , a calibration u α with period α in C α . A 1-calibration on a manifold is a closed one-form u obeying |u| ≤ 1. The new program, called the primal (since there will be a dual program below), is also convex and takes the form Here m α is an arbitrarily chosen representative of C α . There are no conditions on the calibrations at the boundary of M, if it exists. Any feasible u α , provides a rigorous upper bound on the value of the minimum. We can solve the period constraints on the calibrations u α using a basis of real closed one-forms ω i with periods ω i α ≡ C α ω i . We then write u α as as a linear combination of the basis one-forms plus an exact one-form dφ α where φ α is a function on M: The constants c α i are constrained by the period conditions C α u α = α : Having replaced the calibrations by the constants c α i and the functions φ α , where the zero mode of φ α drops out, and the program reads: In [2] we gave several derivations of a maximization convex program that is equivalent to the above primal program. One of the derivations simply applies Lagrangian duality to the primal, hence we term it the dual minimal-area program. The variables are, for each homology class C α , a function ϕ α on the surface and a constant ν α . Choosing a set m α of representatives in C α the program takes the form Dual MAP: (2.7) As stated above, the function ϕ α has discontinuity −ν α across m α . Moreover, M = M\ ∪ α∈J m α is the manifold M with the chosenrepresentative curves m α removed. This is just a way to tell us that in calculating the objective with our discontinuous ϕ α there is no delta-function contribution in dϕ α on m α . In trying to maximize the objective the first term tries to make ν α large; however, a non-zero jump forces ϕ α to have a non-zero gradient somewhere, making the second term, which is always negative, larger. The former is linear while the latter is quadratic, so we expect there to exist a maximum.
Any trial values for ν α , ϕ α give a rigorous lower bound on the maximum. Using both the primal (2.6) and the dual (2.7), we can thus bound the solution both above and below. Strong duality holds for this problem: the maximum in the dual coincides with the minimum in the primal; this is the extremal area A. Note also that we don't need to be careful about the orientation of m α or the sign of the jump, since the objective is invariant under ϕ α → −ϕ α . A special case occurs when, for some α, a subset m α of the boundary ∂ M is a representative of the class C α . In this case the representative m α where ϕ α jumps can be chosen to be m α . Since ϕ α is supposed to vanish at the boundary, placing the discontinuity at m α effectively sets the value of ϕ α on m α equal to ν α . The value of ϕ α on the rest of the ∂ M remains zero: A representative of C α that has length α saturates the length condition and must be locally length-minimizing, else there exist representatives that violate the length condition. We call a C α representative of length α an α-geodesic. The location of the α-geodesics can be readily extracted from any solution to either the primal or the dual program. An α-geodesic in the metric g 0 is calibrated by u α : the constraint |u α | 2 0 ≤ is saturated on that curve and furthermore the vectorû α associated to the form u α is tangent to the curve. Thus the α-geodesics are the closed integral curves of u α on which |u α | 2 0 = everywhere. In a solution to the dual program, wherever dϕ α = 0, the curves of constant ϕ α are α-geodesics. This implies that u α is related to dϕ α (where dϕ α = 0) by (the sign is by convention). One can also show that the Weyl factor of the metric is and since we write = ρ 2 , we have ρ = β |dϕ β | 0 . Moreover In other words, the area for the extremal metric equals the total area of a set of flat rectangles of height ν α and length α . Equation (2.10) is also a powerful constraint on the solution and can be rewritten as a sum rule α |dϕ α | = 1, at every point on M. (2.12) As explained in [2] we can construct Gaussian normal coordinates using a single band of systolic geodesics in the class C α . We use a function x α that parameterizes the α-geodesics by length as the first coordinate. The calibration u α can be identified The second coordinate is ϕ α which, as required, is constant along the systolic geodesics (Fig. 5). The cotangent metric g −1 is constrained by the conditions |dx α | = 1 and dx α , dϕ α = 0: This is the metric in Gaussian normal coordinates. As a consequence of the sum rule (2.12), which now reads α: dϕ α =0 |h α | = 1, in a region where there is a single set of systolic geodesics |h| = 1 and the metric must be flat. If there is more than one set of systolic geodesics then the metric can have curvature. It is a simple consequence of the Gauss-Bonnet theorem that over a band of closed systolic geodesics the integral of the Gaussian curvature must give zero. This follows because the band is topologically an annulus of zero Euler number and the boundaries of the annulus are geodesics. On a band R α of geodesics in the class C α we can use the metric (2.13) to define the height b α (x) of the band as a function of the position x α along the band: (2.14) Assume now there is a value x α 0 of x α for which the integral is done over a region where R α is the only systolic band. Then |h α | = 1 along this segment and This gives simple characterization of ν α : it is the height of the band in a region of the surface where it is the only band.

The Torus with a Boundary and the Associated Swiss Cross
In this section we show how a Swiss cross defines uniquely a torus with a boundary, and vice versa. We then formulate a minimal area problem on the Swiss cross and a minimal area problem on the torus with a boundary. We show that the extremal metric is the same for both problems. Consider first a square torus with a boundary. The square torus, a torus with modular parameter τ = i, is defined by a square region in the complex plane with opposite edges identified. We will take the length of the edges to be s > 0, a fixed quantity. The torus also has a boundary, obtained by removing a square region with side length s − a, where 0 < a < s . The torus square region and the boundary square are aligned and have a common center. The result is shown in the upper left of Fig. 6. We also show in this figure two closed curves, γ 1 and γ 2 that are representatives of the homology classes C 1 and C 2 , respectively. With s fixed one expects different values of a to represent conformally inequivalent tori with boundary. When a → s we have a square torus with a boundary that is becoming so small that is turning into a puncture. When a → 0 the boundary is becoming as big as possible-in string field theory this would correspond to an open string diagram with very long open-string propagators.
We now show an alternative presentation of the surface. For this purpose we cut the surface into four regions denoted as 1, 2, 3, and 4, as shown in the upper right of Fig. 6. The four regions are glued back as indicated on the lower left, and we obtain a "Swiss cross" presentation where horizontal and vertical dashed edges must be identified to form a torus. The Swiss-cross presentation is shown again in the bottom right, with identifications indicated by similar arrows, and with the closed curves γ 1 and γ 2 displayed. We call this a toroidal Swiss cross and it is exactly the same Riemann surface as the torus with a boundary, only the presentation is different.
We are interested in the metric of minimal area on this surface with the condition that all curves in the homology classes C 1 and C 2 , with representatives γ 1 and γ 2 , respectively, have length greater than or equal to s . With this choice of systole we see that the constant metric ρ = 1 is an admissible metric. We will call this the torus problem. One can discuss this problem in the original presentation or in the presentation as a "toroidal Swiss cross", as the surfaces are the same. The minimal area metric should be invariant under all the discrete transformations that leave the torus invariant. This follows from strict convexity of the area functional: if the extremal metric and its transformed version are different, the average metric would have lower area while satisfying all the constraints. The general version of this argument was given in [2], at the end of section 3.1.
We also want to introduce the conventional Swiss cross, or just Swiss cross. This is a planar surface, with shape as shown at bottom right of Fig. 6. In the Swiss cross there are no identifications and the figure is conformally equivalent to the unit disk, with the twelve turning points of the cross resulting in special points on the boundary of the disk. We will call "vertical edges" and "horizontal edges" the segments that were identified in the toroidal Swiss cross.
It is clear from the construction in Fig. 6 that given a square torus with a centered square boundary we can construct the associated Swiss cross. Moreover, given a Swiss cross, by identification of the two vertical edges and identification of the two horizontal edges we obtain a square torus with a centered square boundary.

The minimal-area problems on the Swiss cross and the torus.
We have already stated the torus problem, namely, the minimal-area problem on the torus with a boundary, or equivalently, on the toroidal Swiss cross. Since it has no identifications, on a Swiss cross there are actually two minimal-area problems that could be posed. Each uses a different set of curves to be constrained: 1. Any curve beginning on the left vertical edge and ending on the right vertical edge, and any curve beginning on the bottom horizontal edge and ending on the top horizontal edge. 2. The curves above with the additional condition that beginning and ending points would be identified in the torus version of the Swiss cross.
The set of curves (2) is contained in the set of curves (1). The curves in (2) are the curves that would be closed if we made our Swiss cross toroidal. We define the Swiss cross problem as the minimal area problem that uses (1): the constraints apply to all curves from left to right and all curves from bottom to top.
We now claim that: The extremal metric for (2) is extremal for (1). Namely, constraining the would-be closed curves is enough to make sure all curves are good. To see this, work by contradiction. Suppose we have an extremal metric for (2) in which there is a curve γ 0 in (1) that is too short. Let γ 0 be a curve joining the vertical edges, as shown in Fig. 7. Reflection about the vertical axis going through the center of the cross is an isometry of the extremal metric. Therefore, this reflection gives us another curve γ 0 of the same length as γ 0 and intersecting γ 0 on the axis. This cuts γ 0 into two pieces of lengths a and b with a + b < s , because γ 0 is too short. The length of the pieces of γ 0 are also a and b. We can then use a piece of each curve to form two curves of type (2), one of length 2a and one of length 2b. Whether a < b or a > b, or a = b at least one of the two curves is shorter than s . This is a contradiction, showing that the extremal metric for (2) satisfies all the constraints (1) and is thus extremal for (1).
Consider now a square torus with a boundary and its associated Swiss cross. We will show below that the extremal metric for the Swiss cross problem provides an extremal metric for the torus problem. Claim: The extremal metric for the Swiss cross problem is also extremal for the torus problem.
Proof. Consider the Swiss cross with its extremal metric and copy the metric onto the torus with a boundary. Let the area of (either) surface with this metric be A 0 . We first show that the resulting metric on the torus is admissible for the torus problem.
Consider the curves on the torus in the class C 1 . As they are drawn on the Swiss cross they take two possible forms, as illustrated in Fig. 8. One type of curve, exemplified by γ 1 is simply a curve going from the left edge to the right edge, and is certainly long enough. The second type of curve is not a continuous curve on the Swiss cross and is exemplified by γ 2 , which is homologous to γ 1 . This curve, while going from left to right on the cross, would be continuous if we used the identifications of the top and bottom edges, as shown in the figure to the right. Because of the isometry, however, such a curve has the same length as the curve γ 2 shown below, where we moved a piece of the curve to make it continuous on the cross. As shown, γ 2 is a standard curve going from left to right and must be long enough, showing that γ 2 is long enough. Any curve of the second type can be shown (using the isometries) to have the same length as a curve that goes from left to right. Since both types of curves work and exactly the same argument works for the curves in the class C 2 , the metric induced on the torus is indeed admissible. Now assume the extremal metric on the torus has area A 1 < A 0 . Being extremal, the metric must have the isometries of the torus. That metric copied into the Swiss cross would also have area A 1 < A 0 and the isometries of the Swiss cross. Moreover, since The curves γ 1 and γ 2 on the torus are shown on the Swiss cross. γ 1 is a standard curve on the Swiss cross but γ 2 is not continuous. The curve γ 2 of length equal to that of γ 2 is continuous and standard this metric is admissible for the torus the would-be closed curves of the Swiss cross would be good. Having the isometries, this metric would solve also the Swiss cross problem, in contradiction with our original statement that the minimal-area metric on the Swiss cross had area A 0 .

The Programs for the Torus with a Boundary
We formulate the primal program for the torus with a boundary. We use symmetry arguments to simplify the ansatz for the calibrations and reduce the work to a fundamental domain one-fourth the size of the original surface. We then do a similar analysis for the dual program, taking our time to obtain a lower bound for the extremal area using a simple ansatz for the relevant ϕ functions. We conclude by finding conformal maps of the fundamental domain in which the corner point is mapped to a regular point on the boundary (the pentagon frame) or is mapped to infinity (the strip frame).

The primal program.
We now set up the program for the torus with a boundary, presented as a Swiss cross with identified edges, as shown in Fig. 9. Let us work in the convention where a = 1 and the horizontal (and vertical) parameter length s is also the systole. We must deal with two homology classes: the class C 1 corresponding to curves traveling horizontally and the class C 2 corresponding to curves traveling vertically. Moreover in this surface we have a basis of real closed one-forms spanned by dx and dy. We consider now two calibrations, that is two closed one-forms u 1 and u 2 , corresponding to the classes C 1 and C 2 respectively. Following (2.4) we write the following ansatz for the calibrations where φ 1 and φ 2 are functions on the torus that we have to determine and c 1 , c 2 , c 1 , c 2 are constants also to be determined. The exact one-forms dφ 1 and dφ 2 do not contribute to integrals over closed cycles and therefore the period conditions determine c 1 = 1 and c 2 = 1 so that We now use the symmetries of the surface to constrain further the undetermined coefficients and the form of the functions φ 1 and φ 2 . Locating the center of the toroidal Swiss cross at the origin of the (x, y) plane we have a Z 2 × Z 2 symmetry generated by the transformations The first Z 2 flips the sign of x and the second Z 2 flips the sign of y. The extremal metric must be invariant under these symmetries and therefore the calibrations, which constrain the metric via ≥ |u α | 2 0 , will transform into themselves up to signs. For an arbitrary object χ we write with 1 , 2 equal to plus or minus one, if χ transforms as χ → 1 χ under the first Z 2 and as χ → 2 χ under the second Z 2 . Since u 1 contains dx and u 2 contains dy we must have It now follows from this condition that c 2 = c 1 = 0 and therefore The symmetries constrain the derivatives of the functions φ 1 and φ 2 . Writing the transformations (4.6) now imply that We will see shortly that such results are consistent with our expectations. There are additional symmetries of the surface, generated by a counterclockwise rotation R of 90 • . This acts as R : (x, y) → (−y, x). Composing this with the first Z 2 we have a symmetry Under this symmetry we have that R exchanges the calibrations: One quickly verifies that this implies that A short calculation using these relations and integrating the functions starting from the origin shows that This means that, up to a constant, φ 2 is determined by φ 1 . This is reasonable; one could hope for the simpler R : φ 1 → φ 2 but the constant ambiguity is present because both φ 1 and φ 2 appear acted by exterior derivatives. We can learn a bit more about the functions φ 1 and φ 2 from the symmetry conditions. Consider first the vertical edges.
relating two points on the vertical edge that are identified. Thus ∂ y φ 1 = 0 on the vertical edges and φ 1 is constant along the vertical edges. Since the value of φ 1 over the surface can be changed by an overall constant without any effect, we can choose to make φ 1 vanish at the vertical edges. An identical argument holds for φ 2 and we thus conclude that φ 1 = 0 at the vertical edges, φ 2 = 0 at the horizontal edges. (4.14) In particular the value of φ 1 on the y axis is zero. Pick now two points on the y axis related by reflection about the x axis. Since ∂ x φ 1 is even under y flips, integrating horizontally ∂ x φ 1 from these two points shows that φ 1 is even under y flips. All in all, and including the analogous results for φ 2 , we have φ 1 ∼ (−, +) and is zero on vertical edges and they axis, and is zero on horizontal edges and the x axis. (4.15) This means that we can use as a fundamental domain the part of the surface lying in the first quadrant x, y ≥ 0. If φ 1 and φ 2 are known there, they can be extended accordingly to the full surface. The region is shown in Fig. 10. Finally, noting that (4.15) implies that φ 1 (0, 0) = φ 2 (0, 0) = 0 we now have that (4.13) gives us the simple relation If we know φ 1 on the fundamental domain, we know φ 2 as well. Effectively we just have one unknown and it can be taken to be φ 1 .
The fiducial metric g 0 μν on the surface is taken to be the constant unit metric: The program, using (2.6) then becomes Here the fiducial norms are easily calculated: At this point the program is explicitly defined. Any trial functions φ 1 and φ 2 on the torus will give an upper bound on the minimal area. Indeed, for any choice of φ 1 and φ 2 we compute at every point on the surface the two norms u 1 | 2 0 and u 2 | 2 0 and set equal to largest one. We integrate over the surface and that gives the bound. In searching for the extremal metric, however, we can use the symmetry constraints discussed above and restrict ourselves to the fundamental domain.
We can give some intuition for the constraints we derived above. Consider again the Swiss cross in Fig. 9. The Z 2 × Z 2 invariance of the extremal metric implies that systolic geodesics in C 1 are invariant under reflections about the y axis and systolic geodesics in C 2 are invariant under reflections about the x axis. Additionally, reflections about the x axis map systolic geodesics in C 1 to systolic geodesics in C 1 and reflections about the y axis map systolic geodesics in C 2 to systolic geodesics in C 2 . The systolic geodesics must hit orthogonally the edges that are to be identified, otherwise the reflection invariance would imply a kink on the geodesic at that point.
As stated in Sect. 2, on a systolic geodesic in the class C, with calibration u, the geodesic is tangent to the vectorû obtained from the one-form u using the metric. The components of the vector are then meaning that the tangent direction is simply defined by (u x , u y ). For the saturating geodesics in C 1 the tangent direction iŝ At the vertical edges the previously derived condition ∂ y φ 1 = 0 means the systolic geodesics hit the vertical edges orthogonally, as expected. Looking at the tangents to saturating geodesics along C 1 is is clear that ∂ x φ 1 is even under reflections about the x axis and the y axis, while ∂ y φ 1 is odd under reflections about the x axis and the y axis. This is exactly as anticipated on the first line of (4.9). We have discussed the ansatz for the calibrations and how symmetries simplify the construction of the program. A numerical solution, however, requires discretization. For the benefit of interested readers, we present those details in "Appendix A".

The dual program.
We now consider the use of our dual program to study the torus with a boundary. In this section, as in the previous one, we will set up the program for the case of a torus with systole s and a = 1. This corresponds to the Swiss cross of area 2 s − 1 for the fiducial metric ds 2 = dx 2 + dy 2 . The torus, shown in Fig. 11 as a region on the (x, y) plane, displays a horizontal curve in the homology class C 1 and a vertical curve in the homology class C 2 . We are constraining the length of closed curves in these homology classes.
Having two homology classes the program requires two functions ϕ 1 and ϕ 2 defined all over the torus. Following the conditions in the dual program (2.7) these functions Fig. 11. The torus with a boundary with the elements needed for the dual program. Shown are curves in the homology classes C 1 and C 2 . The function ϕ 1 has a discontinuity across m 1 and the function ϕ 2 has a discontinuity across m 2 ϕ α (α = 1, 2) must have discontinuities ν α across some arbitrarily selected curves Due to the symmetry of the square torus and since the two functions are on the same footing, their discontinuities must be exactly the same: We can set up the discontinuous functions symmetrically: as y → 0 + on the horizontal line m 1 we take ϕ 1 → ν/2 while as y → − s on the copy of m 1 we have ϕ 1 → −ν/2. Doing similarly for ϕ 2 we have: These values are indicated on Fig. 11. As noted in Sect. 2, we need not be careful about the orientation of the homology cycles nor about the sign of the jump. Since we have a torus, the two functions also satisfy the periodicity conditions Apart from the discontinuities, as stated in the program (2.7) the functions must vanish on the boundary of the surface, thus where ∂ M, shown in Fig. 11, is the inner square. The dual program (2.7) applied to our torus with a boundary becomes subject to the discontinuity conditions, periodicity conditions, and boundary conditions given in (4.23), (4.24), and (4.25), respectively. For any set of trial functions ϕ 1 and ϕ 2 satisfying these conditions the above maximum gives a lower bound for the minimal area.
In searching for the optimum, we can use symmetry to constrain the functions. First, the symmetry (x, y) → (y, x) of the torus implies that one function is determined in terms of the other: Indeed, the values of ϕ 1 along m 1 make this the only possible choice. The antisymmetry implies that ϕ 1 vanishes for y = s /2: Given these symmetries, the program can be formulated restricting the torus to a fundamental domain one-fourth the size of the surface and shown shaded in Fig. 11. As for the primal, we discretize the fundamental domain and define the function ϕ 1 on all lattice points, using the discontinuity conditions and the boundary conditions. The function ϕ 2 can then be read from the values of ϕ 1 . At the center of each plaquette one can evaluate the fiducial norms |dϕ 1 | 0 and |dϕ 2 | 0 . The objective is now constructed by adding the contributions from all plaquettes and the optimum is found by searching over positive values of ν and all functions ϕ 1 with the associated discontinuity. To illustrate the formalism and obtain a lower bound on the minimal area, we evaluate the objective using a simple ansatz for ϕ 1 when the discontinuity ν is given. For this we focus on the fundamental domain shown in Fig. 12a. Consistent with the indicated boundary conditions we construct a function ϕ 1 , which along with ϕ 2 , is plotted in Fig. 13. These functions are (4.30) It follows that (4.31) Introducing the subregions D 1 , D 2 , and D 3 shown on Fig. 12b, we have (4.32) We can now evaluate  This means that the minimal area A on the torus with a boundary is constrained to obey For s = 3 one recovers the familiar lower bound A ≥ 9 2 [18]. When the dual program is solved exactly and we have an optimum the systolic geodesics are identified as the lines of constant ϕ α in regions where dϕ α = 0. We can look at these curves for the optimum of our ansatz. We have dϕ 1 = 0 in the region D 1 ∪ D 2 , and dϕ 2 = 0 in the region D 1 ∪ D 3 . The lines of constant for ϕ 1 and ϕ 2 (over the regions where their gradients are non vanishing on the full torus) are shown in Fig. 14a. The region D 1 is covered by two sets of curves, the regions D 2 and D 3 by a single set of curves. The associated metric can also be read from (2.10) and reads (4.37) Recalling that = ρ 2 where ds = ρ|dz|, this is a two-constant metric: ρ 1 = 2ν * on region D 1 and ρ 2 = ν * in regions D 2 and D 3 . We see that the metric is higher in the region with two sets of constant ϕ curves and lower in the regions with one set of constant ϕ curves. Indeed, it makes sense to raise the metric in the region where more geodesics can gain length. This two-constant metric (see Fig. 14b) was obtained by Marnerides [18] who used Beurling's criterion to show that it is the minimal-area metric for a problem where we only constrain the strictly horizontal and strictly vertical curves on the torus. One can easily check that on this metric all these curves, shown in Fig. 14a have length equal to s . Of course, s is not the systole for this metric, as there are non-contractible curves shorter than s . One such curve γ is shown in Fig. 14b: making use of the discontinuity of the metric, it bends slightly to avoid the region where the metric is ρ 1 .

Using other conformal frames.
As it will become clear in the following section, the form of the optimal metric near and at the corners of the central square of the Swiss cross is difficult to assess from the numerical data. While the metric seems to diverge as one approaches these 'central corners', the behavior of the curvature is less clear. As we travel along the Swiss cross in the counterclockwise direction, the central corners are points where there is a turning angle of −π/2. The turning angle is defined as that rotation angle angle of magnitude less than or equal to π taking the tangent to the curve right before the critical point to the tangent to the curve right after the critical point. The rotation angle is measured as positive in the counterclockwise direction and as negative in the clockwise direction. As opposed to the central corners, the other corners of the Swiss cross all have turning angles of π/2.
In this subsection we give two alternative presentations for the Swiss cross, for convenience chosen to have s = 2. They are both conformal maps of a portion of the Swiss cross, an L-shaped region with a single central corner, into polygons where the central corner has become an ordinary point or a point at infinity. These polygons are chosen because they are amenable to discretization. We will give two presentations. One is a pentagon with the central corner mapped to the middle of one edge, and the other is a semi-infinite strip with the central corner pushed to infinity.

Pentagon
Let us consider first the pentagon presentation. As we will see in the next section, the fact that this frame straightens out the boundary at the corner greatly clarifies the behavior of the metric in the vicinity of this point.
In Fig. 15, top left, we show the L-shaped subregion of the Swiss cross as a domain in a w plane (with the full Swiss cross in the little figure above). The pentagonal region, shown to the right as a domain in thew plane, is expected to be conformally equivalent to the L-region for some choice of the parameter α. In the conformal map corners labeled with the same letters go into each other. Our goal is to find the value of α. We will demonstrate the map exists and calculate α by mapping both shapes into the same z upper-half plane with some marked points.
We first map the L-shaped region in the w plane into the z upper half plane with the central corner P mapped to z = 0, the R, R corners mapped to ±1, the S, S corners mapped to ±λ, and Q mapped to infinity. Using the standard rules for turning angles in the Schwarz-Christoffel map, we have where N and λ are unknown parameters. The value of λ is determined by the condition that on the L-shaped region the side P R (mapped to z ∈ [0, 1]) has the same length as the side RS (mapped to z ∈ [1, λ]). This condition gives Given our purposes a numerical solution for λ is all we need. We find, to excellent accuracy, λ = 1.154 701. (4.40) With this value the ratio of the length of the two edges differs from one by less than 10 −6 . The next step is mapping the pentagon in thew plane to the upper half plane as well. Again, we must map the image P of the central corner to z = 0, the R, R corners to ±1, the S, S corners to ±λ and Q mapped to infinity. This time the conformal map takes the form For the L region and the pentagon to be conformally equivalent the value of λ here must be that in (4.40). Note that, as required, there is no turning point at z = 0, corresponding to the central corner. The unknown in the pentagon is the length α of the ST and S R edges. By simple geometry we find that the length of the P R segment is where the last equality follows by numerical evaluation of the two integrals with the known value of λ. We now find

Strip
Let us now turn to the map of the L-shaped region to a semi-infinite strip. We include this discussion mainly for completeness and because it could be useful in future investigations. The results we present in the rest of the paper were not obtained using the strip frame, so the reader should feel free to skip this subsection.
Our task becomes much easier if we complete the L-shaped region into a square, as shown in Fig. 16, upper left, with a new corner point U . This square can be mapped into the interior of a unit disk |u| ≤ 1, with the four corners and the four edge midpoints mapped into the boundary of the disk in such manner that the symmetries of the square are respected. As shown in Fig. 16, the central corner P ends at the center of the circle, the vertical and horizontal lines passing through the middle of the square become vertical and horizontal lines going through the origin of the unit disk and, finally, because of symmetry, the corners of the square go into points on the unit disk with argument ±45 • and ±135 • . Note that the L-shaped region occupies the interior of the unit disk with arg(u) ∈ [0, 3π 2 ].
Note that in this map the real points T and R on the circle are mapped to plus and minus one, respectively. Moreover, one quickly verifies that the points U and S are mapped to x 1 and −x 1 , respectively, while the points S and Q are mapped to x 2 and −x 2 , respectively, where Having constructed the map from |u| ≤ 1 to the upper half plane, we also need the map from the square into the upper half plane z = f (w). This is a Schwarz-Christoffel map that takes the form , N 0.762 759 78, (4.47) where the constant N was determined from the condition that the square containing the L-shaped region have the correct size. We now have the map z = f (w) from the L-shaped region to the upper half plane, and the map u = g(z) from the upper half plane to the u unit disk. Finally we map that the image of the L-shaped region in the u disk to a semi-infinite strip in the η plane. This is achieved by The central corner at u = 0 has been mapped to the end of the strip, at η = − i 2 + ∞, aligned with the center P of the Swiss cross, at η = − i 2 . The position of the varius additional corners are indicated in Fig. 16.

The Results for the Extremal Metric
We begin this section by discussing the results of the primal and dual programs for a Swiss cross with s = 2 (a = 1). The area can be found to six significant digits. The patterns of systolic geodesics as well as the curvature of the surface are discussed. Special attention is given to the (central) corner points where the extremal metric is found to diverge in a way to make them regular points on the boundary. The global picture of the surface is shown in Fig. 22. We then study the moduli space of the Swiss cross, or torus with boundary, finding that the extremal length conformal invariant λ is, to a good approximation, linearly related to a parameter t describing the shape of the rectangles that define the Swiss cross. In the moduli space we find two critical points. One occurs as the positively curved central dome reaches the edges of the Swiss cross. The second occurs when, in the torus picture, the length of the boundary becomes equal to the systole. The latter one is relevant to the study of the torus with a puncture.

Area, metric, and curvature.
In this subsection, we will explore the extremal metric for the Swiss cross (or equivalently the torus with a boundary) obtained from solving the primal and dual programs described in the previous section. For concreteness, throughout this subsection we will focus on the case s = 2 with a = 1. For this surface, the ρ = 1 metric is admissible and has an area A = 3. We will plot all quantities on a fundamental domain of the Z 2 ×Z 2 symmetry, in both the original frame (referred to as the "L" frame) and the frame in which the L is mapped to a pentagon 5 (referred to as the "pentagon" frame; see Fig. 15). The data we present are from the highest-resolution runs of the primal and dual programs in both frames. Based on the runs in the L frame, we estimate the extremal area A as A = 2.806970(1). (5.1) The quoted error estimate of 10 −6 is derived from the convergence of the results as a function of the lattice resolution; see "Appendix A.2" for details. As explained there, the error appears to decrease like (res) −3 , and the results of the primal and dual are surprisingly close to each other at each value of the resolution. The extremal area is about 6.4% lower than the area A = 3 of the ρ = 1 metric. The systolic geodesics are plotted in Fig. 17. These are obtained as the level sets of the functions ϕ 1 and ϕ 2 of the dual program, and are drawn as blue and red curves, respectively. The function ϕ 2 is plotted in Fig. 18. We see from Fig. 17 that there is a region foliated by 1-geodesics (in light blue), another foliated by 2-geodesics (pink), and a third foliated by both (purple). We will denote the last region as U 2 . If we follow the uppermost 1-geodesic from the left in the L frame, it appears to meet the vertical boundary tangentially, before making a sharp 90 • left turn counterclockwise and and following the horizontal boundary; similarly for the rightmost 2-geodesic. In the pentagon frame, these last geodesics join the diagonal boundary of the pentagon tangentially at its midpoint. Fig. 18. The optimal configuration of the function ϕ 2 appearing in the dual program, drawn in the a L frame and b pentagon frame. The region where the gradient of ϕ 2 is non-zero defines the red and purple regions of Fig. 17. The level sets of ϕ 2 are the 2-geodesics shown in red Next, we study the extremal metric, or more precisely its line element ρ = √ . Figure 19 shows ρ, ln ρ, and a contour plot of ln ρ in both the L and pentagon frames. Of course, ρ is not a scalar, so it gets rescaled by the factor |dw/dw| between the two frames, where w is the L coordinate andw is the pentagon coordinate (see Fig. 15): In the L frame, ρ exhibits a spike at the inner corner; however, it remains finite at the corresponding point in the pentagon frame, the center of the diagonal boundary (labelled P in Fig. 15). Near that point, the conformal map takes the form where w P andw P are the wandw-coordinates of P respectively. Hence Given the finiteness of ρw near P, equation (5.2) gives the following prediction for the behavior of ρ w near P: To check that our ρ w is consistent with the prediction (5.5), in Fig. 20 we plot the difference ln ρ w − ln(|w − w P | −1/3 ); indeed, it is clearly bounded near the corner. We can say that the minimal-area problem wants the boundary to be smooth; by giving the metric the spike (5.5), it straightens out the corner that is present in the L coordinate system. The same effect leads to the spikes in ρw at the two corners of the pentagon. The pentagon is the Z 2 ×Z 2 fundamental domain of a torus with a diamond (i.e. a square with sides at 45 • ) removed. The corners of the diamond are at the points labelled R and R in Fig. 15. To straighten out those corners, the minimal-area problem makes the metric diverge like |w −w R | −1/3 and |w −w R | −1/3 . On the other hand, the boundary of the torus is already smooth at the points R, R in the L frame, so the metric remains bounded there. In a coordinate system in which the boundary has no corners-for example, a torus with a disk removed-the metric would be bounded everywhere.
Another notable feature of ρ in both frames, apparent in the plots of Fig. 19, is the "dome" occupying the region U 2 and bounded by a crease coinciding with the rightmost and uppermost systolic geodesics. We compute the Gaussian curvature K using the formula where ∇ 2 0 is the Laplacian in the fiducial metric. We find that K is positive in U 2 , increases as we approach the boundary of U 2 , and has a negative delta-function along its boundary, indicating a line curvature singularity. This can be confirmed by computing K numerically, as shown in Fig. 21. Our numerical results seem to indicate that the curvature K remains bounded as the boundary of U 2 is approached; however, we cannot make a definitive claim about this. We also see, as expected, that the curvature K vanishes in the regions foliated by just one band of geodesics.
Based on this data, we now put together a global picture of the minimal-area metric on the Swiss cross. For this purpose, we return to the full Swiss cross, not just one Z 2 × Z 2 fundamental domain. It consists of a positively-curved dome glued to four flat wings. Each wing is a rectangle with a half-disk removed. As can be seen in Fig. 17b, the curves that bound the dome U 2 are tangent where they meet. (In the L frame shown in Fig. 17a, they appear to meet at a right angle; however, as emphasized above, the metric is singular at the corner where they meet, and we want a picture where the metric does not diverge in this region.) There are four such curves, and together they form a complete geodesic circle. Therefore, by the Gauss-Bonnet theorem, the Gaussian curvature integrates to 2π over the dome; the dome is essentially a hemisphere, albeit not with constant positive curvature. There is a delta-function of negative curvature along the curve where the dome is glued to each wing. This curve has different extrinsic curvatures on its two sides; on the dome side it has vanishing extrinsic curvature while on the wing side it has negative extrinsic curvature, with total turning angle π (like a semicircle). Therefore the Gaussian curvature integrates to −π over each gluing curve. This is consistent with the fact that the total Gaussian curvature over any of the two bands of systolic geodesics vanishes, since each such band crosses the full dome as well as two gluing curves. An embedding picture of the whole Swiss cross is shown in Fig. 22. This is not a quantitatively accurate picture of the metric, but shows the important qualitative features. In particular, the dome is pictured as a hemisphere with four points on its boundary "pinched in" to leave 90 • concave angles, allowing the wings to be glued in at the corners; again, the boundary is regular at these corners.
We can understand the qualitative features of the solution as follows. The (flat) fiducial metric just barely obeys the systolic constraints, as the horizontal and vertical line segments connecting opposite edges have length 2. In order for the area to be reduced on any part of the Swiss cross, therefore, it needs to be increased somewhere else in order not to violate the systolic constraint. As can be seen in Fig. 19, the solution achieves this   Thus, while naively it may appear from Fig. 22 that the area could be decreased further by shrinking the dome, in fact that would violate the systolic constraint.

Moduli space of the torus with a boundary.
In this section we investigate Swiss crosses-or tori with a boundary-of different shapes. We identify a simple parameter, called t, that specifies the shape, and attempt to relate it to the extremal length conformal invariant λ that is a simple function of the extremal area. Our numerical analysis suggests a rather surprising linear relation between λ and t, but careful consideration shows that the relation does not hold exactly. We also show how one can use the extremal metric on one surface to construct, by simple amputation, the extremal metric on a surface of different shape.
We suggested in Sect. 3.1 that with fixed s the parameter a would be a modulus. In the Swiss cross presentation of the torus (Fig. 6), the parameter a is the length of the vertical and the horizontal edges that are identified. Alternatively, we can think of the Swiss cross as the part of the z plane covered by two rectangles, each of length s and width a, one placed horizontally and the other vertically, with coincident centers, as shown in Fig. 23.
Clearly, for fixed s we can take a ∈ (0, s ). As a → 0 the boundary of the torus grows until it eats up the whole torus. In the limit a → s the boundary of the torus is becoming infinitesimally small. Alternatively, for fixed a we can take s ∈ (a, ∞).
We can use the modulus of each of the rectangles as a tentative modulus for the torus with a boundary (or the Swiss cross). The modulus of a rectangle is the ratio of the two sides. We thus define the modulus t: t is a modulus if we can find a conformal invariant λ of the surface such that t = t implies λ = λ . As we explain now, the requisite conformal invariant, called extremal length, can be constructed using data from the minimal-area metric.
The extremal length λ [25] is defined for a surface with a set of curves : For a given choice of metric ρ the length L( ) is the length of the shortest curve γ in the set , and A(ρ) is the area of the surface in that metric. The ratio L 2 ( )/A(ρ) is manifestly invariant under scaling of the metric by an arbitrary constant. After computing this ratio one searches over all possible metrics to find the largest possible value for the ratio. The extremal length is determined by the minimal area metric as follows. Each metric ρ can be rescaled so that L( ) = s , where s is some arbitrary length, now the length of the shortest curve in . We then have Since the supremum is achieved by having A(ρ) as small as possible, λ is the ratio of s -squared divided by the area A of the least-area metric for which the shortest curve in is longer than or equal to s : For our problem the surface is the torus with a boundary and is the set of curves in the two homology classes specified earlier. Since the shape of the torus with a boundary is fully determined by the parameter t, the conformal invariant λ must ultimately be a function of t: For a torus defined by some value of t, the area A(t) of the minimal-area metric determines the extremal length λ(t). Recall that for the extremal metric we also have, from (2.11), A = 2 s ν, (5.12) where ν is the optimal common parameter for the bands of geodesics. It is also, as explained around (2.15), the length in the extremal metric of the terminal edges of the Swiss cross presentation, where there is only one foliation. These edges have parameter length a in our construction of the surface. From the last two equations the conformal invariant λ(t) can be given in terms of ν: With ν the length of the edge a of the Swiss cross, we expect ν → a as s → ∞ because ρ → 1 on the arms of the cross. Our goal now is to determine the function λ(t) for t ∈ (1, ∞). We will see that λ is a monotonic function of t, meaning that t is indeed a modulus. Moreover, we find that to first approximation the relation between λ and t is surprisingly simple. This relation, however, is not exact.
To determine λ(t) we solve for the extremal area for tori with several values of t and then attempt to guess a relationship. For these calculations we take a = 1 and vary s . As a result a = 1 → t = s , (5.14) the modulus equals the systole. In the table below we computed the extremal area A and the modulus λ for integer values of t from 2 to 11.  The above data is shown in Fig. 24. A linear fit seems appropriate and gives λ(t) 0.425052 + 0.50001 t. (5.15) This led us to ask: Can it be that we have an exact relation of the form Here λ 0 would be a constant with value of about 0.425. First note that this cannot hold for all t ∈ [1, ∞]. As t → 1 the Swiss cross becomes a square with side length equal to one, the minimal-area metric is ρ = 1 and the extremal length λ(1) = 1. This is certainly not consistent with (5.16). Still, it is conceivable that for t greater than some critical value t 0 a linear relation could be exact. The critical value t 0 1.73 is such that for t > t 0 the edges of the Swiss cross lie beyond the region with curvature. Even for t > t 0 we believe that (5.16) is not exact. Numerical exploration, which we describe now, suggests that with the assumption of a linear relation, the slope is close but not exactly equal to 1/2. where the absolute error | | is estimated from the convergence rate as the lattice size is increased. In fact, the difference between the above value and the value obtained from The estimate here is rather conservative, the error is probably less than 1 × 10 −5 ; indeed the difference between the above value for the area and the value obtained from the primal is 6 × 10 −7 . Given this information we get λ(3) − λ(2) = 0.500 05 ± 0.000 02. (5.19) This is not consistent with λ(3) − λ(2) = 0.5, as would be predicted by (5.16), thus making clear that such a simple linear relation between λ and t does not hold. We have learned that the minimal-area metric on the Swiss cross is positively curved on the region with a double foliation and flat in the four regions with a single foliation (see Fig. 3). The two types of regions are separated by negative line curvature singularity. Let us focus on the regions with flat metric. Having just one band of geodesics the metric in that region can be written as (5.20) and the region (stretching up to the right vertical edge of the Swiss cross) is shown to the left in Fig. 25. The region is a piece of a rectangle: the bottom and top horizontal boundaries are the curves ϕ = 0 and ϕ = ν, respectively and the geodesics are the (horizontal) curves of constant ϕ. All the geodesics are orthogonal to the edge, which must then be a (vertical) line of constant x. In this region, where we can define the complex coordinate w = x + iϕ, we have ρ = 1. In the z complex plane where the Swiss cross is defined, the numerical work indicates that the metric on the flat region is not constant. This means that the map between the z-plane flat region and the w-plane flat region of Fig. 25 is nontrivial. The presentation using the w plane, which can be used separately for each of the four flat regions of the Swiss cross, allows us to use amputation to construct extremal metrics for Swiss crosses of different moduli.
Let denote a Swiss cross with extremal length λ and consider the w plane presentation of the flat sides of the surface with the extremal metric. Let denote the surface obtained by cutting, on each of the four w plane presentations, small rectangles of length /2 and height ν, while keeping the metric the same (Fig. 25). The new surface is still a Swiss cross because it is conformally equivalent to a round disk where the edges are invariant under orthogonal reflections.

Claim: The metric induced on
is of minimal area under the condition that all curves between the new vertical edges and a between the new horizontal edges are longer than or equal to s = s − .
Proof. First note that all relevant curves in have length greater than or equal to s because if there was a shorter curve adding back the removed rectangles we could extend that curve to one in the original surface with length less than s , which is impossible. Moreover, if there was a metric on with less area and still having all curves longer than or equal to s we could add back the rectangles and obtain a new metric with less area on .
Using (5.13) the new surface has extremal length It is also clear that the new area is still equal to twice the systole times the edge length: If we know the t parameter of the initial Swiss cross , however, the cutting procedure does not tell us the t parameter of . Indeed, since the mapping from the z plane to the w plane is nontrivial, the w-plane cuts do not correspond to straight cuts in the z-plane presentation and thus the new value of the parameter t cannot be identified.

Special tori in the moduli space.
Setting a = 1, the torus with a boundary, or Swiss cross, is determined by the parameter s , which we take to be equal to the systole in the minimal-area problem. There is another parameterization of the various surfaces that is useful when we want to keep the systole constant and equal to one. In that case we view the surface as a torus with a square boundary and the parameter, called h, is the length of the edge of the square boundary. The relation between s and h with a = 1 is illustrated in Fig. 26: The Swiss cross with s = 3 corresponds to h = 2/3 while the Swiss cross with s = 2, which we used to describe much of our results, corresponds to h = 1/2, a boundary edge half as large as the systole. In this section we will identify two critical values of h at which the minimal-area metric undergoes interesting changes.
As we have observed with the h = 2/3 and h = 1/2 solutions, the metric in the Swiss cross picture has a central "dome" with positive Gaussian curvature bounded by negative curvature line singularities. The dome is the region covered by two bands of systolic geodesics. As we reduce h further, at some point the line curvature singularities touch the edges of the Swiss cross. This defines a critical value of h that we have estimated at h (1) 0.4201. (5.24) This critical value was found by analyzing the value of the gradient of the ϕ function associated with the geodesics that are about to hit the edge. As we recall, a non-zero gradient at a point implies there is a systolic geodesic at the point. We find that as h approaches its critical value from below, the gradient at the edge midpoint decreases linearly to a very good approximation. Beyond the critical value the gradient becomes  (1) , are shown in Fig. 27. The second critical point occurs for a value h (2) that defines a surface relevant to the minimal area problem of the square torus with a puncture. This problem is discussed in detail in the following section, but we note here a couple of useful facts. In the toruswith-boundary problem studied so far, we have only constrained the lengths of the curves in two homology cycles. The closed curves around the boundary are not constrained. In a theory of closed and open strings, closed curves homotopic to a boundary are also required to satisfy the length condition. Since curves homotopic to the boundary are homologically trivial, our methods require modifications that are explained in the following section. For now assume we can impose this extra condition.
We claim that for h > h (2) , in the minimal-area metric with two length conditions, the length L of the boundary is in fact greater than the systole 1. Even more, all curves homotopic to the boundary are longer than 1. This means that the solution of the two- Fig. 28. Systolic geodesics for the torus (left) or Swiss cross (right) with h = 23/94. This is, to a good approximation, the surface for which the length L of the boundary is in fact equal to the systole 1 condition problem is in fact a solution for the problem with an extra length condition on the curves homotopic to the boundary. As h approaches h (2) from above, L approaches 1 from above. An analysis performed in Sect. 6.2 implies that L = 1 for h = h (2) where Numerical analysis requires a rational representation of h and we can take h = 23/94 0.24468. The geodesics for this torus are shown in Fig. 28. Note that the boundary is surrounded by regions of flat metric with a single band of geodesics. It is curious that h (2) ∼ 1/4, the value for which the flat ρ = 1 metric makes the length of the boundary equal to the systole. For h < h (2) the two-condition minimal area metric has curves homotopic to the boundary that are shorter than 1, and this metric is not a solution for the string field theory problem. If we imposed an additional length condition on curves homotopic to the boundary the minimal-area metric would have larger area and it would have a new foliation by curves homotopic to the boundary. Thus the value h (2) is a critical point in the moduli space: the solution of both problems is the same for h ≥ h (2) and the solution is different for h < h (2) .
More detailed exploration reveals a new feature occurring below the value of the first critical point. We find that for h < h (1) the extremal metric on the two-foliation dome features a region with negative curvature in addition to the region with positive curvature. As the dome collides with itself, the affected portion of its boundary becomes smooth; the line curvature is gone. Negative curvature spreads out becoming bulk curvature. Although a more detailed description will be given in the following section, given that h (2) < h (1) , the surface displayed in Fig. 28 has both positive and negative bulk curvature. The negative curvature in the Swiss cross presentation (right) exists near the four edges, on each one on a neighborhood of the piece of the 'last' systolic geodesic that runs along the edge. In the torus picture these segments are the central vertical curve (red) and the central horizontal curve (blue). The bulk negative curvature exists in some neighborhood of those curves.

The Once-Punctured Torus
In this section we determine the minimal-area metric for a square torus with a puncture. This is perhaps the simplest unknown metric in closed string field theory. This problem is also a natural next step after our study of the torus with a boundary. Indeed, we may view the punctured square torus as the limit of the square torus with a square boundary considered so far, when the size of the boundary goes to zero. For the oncepunctured torus, curves homotopic to the puncture are of nontrivial homotopy and must be constrained in the minimal-area problem of closed string field theory. Such curves, however, are homologically trivial and one cannot have a calibration u with nonzero integral along them. In terms of the conserved flow associated to a closed form, a nonzero integral implies a net flow across any curve surrounding the puncture. But such a flow, sourced at the puncture, has nowhere to go on the surface because there is no possible sink.
We will use a device proposed in [2] to deal with this complication. We consider some covering surface (M, p) of the original surface M, where p :M → M is the projection map. InM the curves surrounding the puncture become homologically nontrivial. The desired calibrations on the covering surfaceM are such that at pointsx that are projected down to x ∈ M the one-form u(x) is equal to u(x) or −u(x). Much of the discussion in [2] was couched in terms of a double cover, which in principle suffices to deal with the situation. In that double cover of the punctured torus, we have two punctures and the curves surrounding the punctures are homologically nontrivial, allowing a calibration to exist. In terms of flows, one puncture can act as a source while the other will act as a sink. We will discuss below the double cover briefly. It turns out, however, that a fourfold cover is far more convenient. The double cover breaks some of the discrete symmetries of the original punctured square torus. On the other hand the fourfold cover breaks none of the symmetries. In the fourfold cover we have a torus with four punctures; two will be sources and two will be sinks. The calibration is indeed the same, up to signs, on points that project to the same point on the original surface M.
We will first discuss the formulation of the programs and then our results.

Programs for the punctured square torus.
For the construction of the requisite oneforms on the double and fourfold cover of the torus, we will use the Weierstrass zeta function ζ(z) defined in terms of the half periods ω 1 and ω 2 of a torus z ∼ z + 2ω 1 and z ∼ z + 2ω 2 . This zeta function is odd and it has a single pole at z = 0 in the fundamental domain of the torus The function is quasi-periodic on the torus: with constants η 1 = ζ(ω 1 ) and η 2 = ζ(ω 2 ). While the dependence of the zeta function on the half periods is usually omitted, some authors use ζ(z; ω 1 , ω 2 ) or ζ(z, g 2 , g 3 ), with g 2 (ω 1 , ω 2 ) and g 3 (ω 1 , ω 2 ) some functions of ω 1 and ω 2 . For the minimal-area analysis of the punctured torus we fix s = 1. In practice, when we discretize, we deal with the puncture by replacing it by a small square hole, thus going back to the problem of a torus with a boundary. The curves surrounding the boundary are constrained and the size of the boundary will be reduced. With the punctured torus being the unit square with identifications, we take the puncture to be at the center λ of the square. In summary: Punctured torus: 2ω 1 = 1, 2ω 2 = i, λ = 1 2 + 1 2 i. (6.4)

Double cover
The double cover is itself a torus with half periods and puncture locations λ 1 and λ 2 as follows: Double cover: 2ω 1 =1, 2ω 2 = 2i, The double cover is shown in Fig. 29. One can imagine cutting open the original punctured torus (center) along a curve that corresponds to the identified horizontal segments at y = 0 and y = i (left). A copy of the punctured torus is glued across those curves to obtain the torus shown on the right and left. Since the punctures must be treated like boundaries, it is clear that in the double cover the curves homotopic to the punctures are, as desired, homologically nontrivial.
To construct the primal we need a calibration for each of the homologies we want to constrain. We have already discussed in Sect. 4.1 the one-forms u 1 and u 2 for curves that run horizontally or vertically on the original torus. These need no modification. We now need to consider a one-form u 3 to constrain the curves that surround the punctures. Following (2.4) we write u 3 on the double torus as follows Here c 1 and c 2 are constants and the last term is an exact form written in terms of some unknown function φ 3 . The closed form ω (3) must have the correct unit period along curves homologous to the puncture P 1 This means that, necessarily For the program one must determine the form ω (3) explicitly, while the constants c 1 , c 2 and the function φ 3 are determined from the minimization of the objective. To find ω (3) we first construct a meromorphic abelian one-form ω dc , with 'dc' standing for double cover. The requisite form, we claim, is given by The first two terms guarantee exact periodicity on the torus: ω dc (z + 1) = ω dc (z + 2i) = ω dc (z). Because it has unit residues at the punctures we have The constant last term in (6.9) ensures the anti-periodicity where z+i and z are related by the projection in the double cover. The form ω dc satisfies all the constraints we want from ω (3) except that it is not a real one-form (a form f dx + gdy with real f and g). The primal is formulated using real forms. This is no complication: since the periods of ω dc are real we can simply set ω (3) equal to the real part of the meromorphic one-form ω dc : Thus defined, ω (3) has the right periods, the proper anti-periodicity and is well defined on the double torus. It is clear in the ansatz (6.6) that c 1 = c 2 = 0 since the forms dx and dy do not have the requisite anti periodicity under y → y + 1. Our ansatz then becomes It is interesting to note that ω dc is in fact the abelian differential that defines a one-loop light-cone diagram with one incoming state and one outgoing state. Indeed, this one-form only has periods around the punctures and around the cycles corresponding to the curves γ 1 and γ 2 of Fig. 29. We can visualize ω dc by plotting its real part h x dx + h y dy as vector field lines with tangent (h x , h y ). This is shown on the left of Fig. 30. To the right we show the associated flow, using the dual form −h y dx + h x dy. The bottom puncture is a sink and the top puncture is a source. The figures make clear that the discrete symmetries of the original punctured square torus are partially broken: neither the form nor the flow, restricted to the original torus, are invariant under rotations by π/2. As we will see now we can get a simpler formulation of the problem using a fourfold cover. We cut open the two cycles corresponding to the opposite edges of the unit square, and glue three copies to form a single square torus double the size, as shown in Fig. 31. The fourfold cover of the punctured square torus is itself a square torus with periods 2 and 2i and four punctures located at the centers λ i , i = 1, · · · 4, of the unit squares: Fourfold cover: 2ω 1 = 2, 2ω 2 = 2i, A closed one-form on this fourfold cover with the correct integrals around the punctures and with the requisite antisymmetry or symmetry under projection to the original manifold is build from a meromorphic abelian differential ω fc , with 'fc' for four-cover: This form has the requisite periodicities ω fc (z + 2) = ω fc (z), ω fc (z + 2i) = ω fc (z), (6.16) as well as the proper anti-periodicities ω fc (z + 1) = −ω fc (z), ω fc (z + i) = −ω fc (z), (6.17) and the required integrals around the punctures, Just as we explained for the case of the double cover, we can use the real part of the meromorphic differential ω fc as our real one form, because it also satisfies all of the requisite properties. The one-form u 3 for the fourfold cover is therefore The abelian differential ω fc is rather special. Its square is a Jenkins-Strebel quadratic differential that defines a one-loop contact interaction of four strings on a torus. Indeed the quadratic differential takes the form ∼ (dz) 2 /(z − λ i ) 2 near each puncture λ i . In Fig. 32 we show the representation of the one-form Re(ω fc ) as vector field lines (left) and the representation of the associated flow (right). The significant advantage of the fourfold cover over the double cover, made manifest in the figure, is that the form ω fc restricted to the original torus has the discrete symmetries of the square. Using the fourfold cover, both in the primal and in the dual programs we can restrict ourselves to fundamental domains within the original punctured torus. In fact, we can focus on the Fig. 32. The meromorphic abelian differential ω fc for the fourfold cover of the punctured torus. Its real part is the one-form represented in this figure. Left: the vector field lines associated with the one-form. Right: the vector field lines for the associated flow. The form and the flow, restricted to the original torus preserve the square symmetry. The abelian differential ω fc , squared, defines a Jenkins-Strebel quadratic differential that corresponds to a contact interaction for a torus and four external states. This is a contact interaction because the union of the development of each of the four strings covers fully the surface Having determined the one-form Re(ω fc ) that provides the unit period around the puncture, let us examine the symmetry constraints on φ 3 . We do this in the context of the regulated puncture, where a small square hole replaces the puncture. The fundamental domain is shown in Fig. 33, left. The integral of dφ 3 around the puncture should provide no extra contribution to the period. It follows that the integral of dφ 3 along any contour starting anywhere on the horizontal edge y = 1/2 and ending anywhere along the vertical edge x = 1/2 must vanish because any nonzero contribution, by symmetry, would give four times that value for the full integral around the puncture. This means that φ 3 must be equal to the same constant on both edges. Without loss of generality we take φ 3 = 0 along these edges, as shown in the figure. A further constraint arises from the Z 2 transformation (x, y) → (y, x). Consider the two points P andP related by such an exchange. The x and y components of the form u 3 at those points must be related as follows u 3 x (P) = −u 3 y (P), u 3 y (P) = −u 3 x (P). (6.20) One can verify that the form Re(ω f c ) has this symmetry. The trivial form dφ 3 must also have the symmetry, so y) . (6.21) These relations require that y). (6.22) With this condition, we now learn that φ 3 = 0 along the diagonal from the origin up to the corner of the small boundary and one can work with the shaded part of the domain (Fig. 33, left). The value of φ 3 is then free on the vertical segment of the small boundary and on the vertical segment x = 0, y ∈ [0, 1/2]. Let us now turn to the dual program. Here we also use the fourfold cover, giving us strong symmetry constraints on the function ϕ 3 associated with the curves homologous to the puncture. Our discussion of the dual implied that the discontinuity ν 3 on the value of ϕ 3 across a representative curve of the homology class can be implemented by setting φ 3 = ν 3 at the boundary of the original torus. On the fourfold cover, the function ϕ 3 must also be anti-periodic under z → z + 1 and under z → z + i. It follows that the value of ϕ 3 at the other three boundaries must be either plus or minus ν 3 . The expected invariance of the systolic geodesics under reflection about the vertical line x = 1 of the covering surface implies that ϕ 3 must be antisymmetric under this reflection. Similarly, ϕ 3 must be antisymmetric under reflection about the horizontal line y = 1. It follows that ϕ 3 = 0 on those lines, and its anti-periodicity implies that ϕ 3 = 0 along the apparent boundaries x = 0, 2, and y = 0, 2, of the fourfold cover. The takeaway of this analysis is shown on the right part of Fig. 33. As indicated there, ϕ 3 = ν 3 on the small boundary, and ϕ 3 = 0 along the vertical segment x = 0, y ∈ [0, 1/2] as well as on the horizontal segment x ∈ [0, 1/2], y = 0. The function ϕ 3 is also symmetric under the diagonal reflection (x, y) → (y, x) and therefore it suffices to find out its value on the shaded region. These are the boundary conditions to be used in maximizing the dual objective.

Results.
We expect the minimal-area metric on the punctured torus to include a semi-infinite cylinder of circumference 1, with the puncture at infinity. It is therefore useful for numerical purposes to cut off the cylinder, by considering instead the square torus with a finite square, of edge size h, removed. If the hole is small enough, it will be surrounded by a region foliated by systolic geodesics that wrap the hole, which we call 3-geodesics. In the minimal-area metric, this region will be a finite cylinder. The rest of the torus will be covered by 1-and 2-geodesics, wrapping the torus horizontally and vertically respectively. Generic 1-and 2-geodesics cannot intersect the 3-geodesics; the reason is that if they intersected they would have to intersect twice, but systolic geodesics may not intersect more than once. The only exceptions are the "last" geodesics of the respective foliations, which may coincide along the mutual boundary of the regions they foliate.
These expectations are borne out by the configurations obtained numerically. Here we present data for the case h = 1/16 from our highest-resolution solution of the dual program (128 lattice points on a side of the fundamental domain). The results from  (2) , defined in Sect. 5.3 as the value of h for which, without imposing the length constraint on curves wrapping the boundary, the boundary has length 1. As argued in the main text, this coincides with h ν 3 =0 , the smallest value of h for which ν 3 vanishes when the length constraint is imposed solving the primal program are closely consistent. The systolic geodesics are shown in Fig. 4. We see four distinct types of regions: The white annular region is foliated by 3-geodesics (shown in black); the two light-blue triangular regions foliated by 1geodesics (blue); the two pink triangular regions foliated by 2-geodesics (red); and the purple region foliated by both 1-and 2-geodesics. The last 3-geodesic coincides with segments of the last 1-and the last 2-geodesic; this is where the cylinder is attached to the rest of the torus.
We obtained the following values for ν α and the area A: The errors on these quantities, as estimated from the difference with the results of a lower-resolution run (64 lattice points on a side of the fundamental domain) are roughly ±1 on the last digit. From the discussion in section 5 of [2], we know that the values of ν α and A should satisfy A = ν α (recall that we have set the systole s to 1), which is indeed satisfied by the values (6.23). Furthermore, since the cylinder is foliated by a single band of geodesics, its height and area are both given by ν 3 . Therefore the rest of the surface must have area ν 1 + ν 2 = 0.91589. (6.24) Before describing the geometry of the h = 1/16 surface, we make a small digression concerning surfaces with different hole sizes. As h increases, the height ν 3 of the cylinder decreases, and at a certain value h = h ν 3 =0 it vanishes (see Fig. 34). For h > h ν 3 =0 , there are no 3-geodesics, and the boundary has circumference greater than 1. Therefore these surfaces are the same as the ones obtained in Sect. 5 without imposing the length constraint on curves wrapping the hole. At h = h ν 3 =0 , the boundary circumference is 1. It follows that h ν 3 =0 = h (2) , the second critical value discussed in Sect. 5.3, and indeed this is consistent with the data obtained for various values of h. On the other hand, for h ≤ h (2) , while the cylinder height depends on h, the geometry of the rest of the  Regions with K > 0 (brown) and K < 0 (gray), with the last 1-and 2-geodesics shown. The complicated pattern in the regions foliated by a single geodesic (top right corner) is presumably due to numerical noise; the curvature of the exact solution must vanish there surface-the complement of the cylinder-should be independent of h up to a complex coordinate transformation. In particular, ν 1,2 should be independent of h. Indeed, we confirmed this prediction for h = 1/8, 3/16, and 7/32, finding values of ν 1,2 equal (within their respective errors) to the one in (6.23). The value of h (2) quoted in (5.25) was found by searching the moduli space of tori with boundary (i.e. without imposing the length constraint on curves wrapping the hole) for the one with ν 1,2 equal to the value in (6.23).
We now return to our study of the geometry of the h = 1/16 surface. The logarithm of line element ρ is plotted in Fig. 35. It stays finite everywhere, and rises roughly linearly in the cylinder region. A crease is clearly visible, which coincides with the boundary of the region foliated by two geodesics. Since the Gaussian curvature is proportional to minus the Laplacian of ln ρ, this crease implies a line curvature singularity with negative curvature, similar to what we saw in the Swiss cross described in Sect. 5.1. Other than Fig. 37. Decomposition of the h = 1/16 torus-with-hole into singly-foliated (left) and doubly-foliated (right) regions. The boundary segments with matching numbers are identified. The singly-foliated region is a cylinder (foliated by 3-geodesics) capped with four "teeth" (two foliated by 1-geodesics and two by 2-geodesics). The doubly-foliated region is a disk, whose boundary is divided into eight segments; four (numbered 5 and 6) are identified pairwise diametrically across the disk, and the other four are identified with the four parts of the boundary of the left figure this crease, the metric appears to be smooth, so we expect the Gaussian curvature K to be finite. K is plotted in Fig. 36. The calculation of K involves second derivatives of the metric; these derivatives, computed on the lattice, greatly amplify numerical noise; this is presumably the origin of the non-zero values for K seen in the regions foliated by a single geodesics, which in the exact continuum solution must be flat. In the region foliated by both 1-and 2-geodesics, K is nearly constant with a value around 1.8 in a region near the center, but then rises in the diagonal directions (toward the hole), and falls in the horizontal and vertical directions, becoming negative near the right and top edges of the fundamental domain. As discussed in Sect. 5.3, this bulk negative curvature is observed for all values of h less than the critical value h (1) where the doubly-foliated region first touches itself (see (5.24)).
We can attempt to illustrate the geometry of this surface as follows. The cylinder is flat. The two pink and two light blue "teeth", foliated only by the 1-and 2-geodesics respectively, are also flat, and join onto the cylinder without any line curvature. This part of the geometry is shown on the left panel of Fig. 37, where the left and right sides of the figure are identified. The doubly-foliated region is shown in the right panel. The boundary of this region is divided into eight segments, four of which are identified with each other pairwise diametrically across the disk and the other four with parts of the boundary of the figure on the left side. The boundary of the disk is a closed geodesic made up of pieces of 1-and 2-geodesics which meet tangently. Since it is topologically a disk, and its boundary has vanishing extrinsic curvature, by the Gauss-Bonnet theorem its total Gaussian curvature must be 2π . It can be thought of roughly as a hemisphere with mostly uniform positive curvature but some "lobes" near its boundary that give it regions of large positive and negative curvature. It has D 4 dihedral symmetry. There is no line curvature where it is identified with itself, since it is identified along geodesics. There is line curvature, however, along the four lines where the left and right sides of Fig. 37 are glued. Each such line is a geodesic from the point of view of the disk (right side) but has total extrinsic curvature −π from the point of view of the "teeth" (left side), so carries total Gaussian curvature −π . The total curvature of the whole surface is thus 2π + 4(−π) = −2π , which of course agrees with that predicted by the Gauss-Bonnet theorem, given that the Euler character of the torus-with-hole is −1.
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A. Numerical Implementation
In this "Appendix" we give some details of the numerical solution of the primal and dual programs for the Swiss cross and punctured torus described in Sects. 4 and 6.1 respectively. For both programs, the fundamental domain was discretized using a lattice, and the derivatives and integrals appearing in the constraints and objectives were replaced by lattice derivatives and integrals. Lowest-order derivatives and integrals were employed, both for simplicity and because-as can be seen in the results presented in Sects. 5.1 and 6.2-the functions being approximated are not necessarily smooth. The discretization scheme in shown in detail in subsection A.1 for the illustrative case of the primal program for the Swiss cross in the L frame; the discretizations for the pentagon frame, the dual program, and the punctured torus are closely analogous.
The minimization and maximization in the two programs respectively was carried out in Mathematica using the built-in functions FindMinimum and FindMaximum. The primal program is a constrained minimization problem, which Mathematica solved with an interior point method. The dual program is an unconstrained maximization problem, which Mathematica solved using a quasi-Newton method with Broyden-Fletcher-Goldfarb-Shanno updates.
In subsection A.2, we compare the results of the primal and dual at various resolutions, focusing on the case of the Swiss cross in the L frame, showing that the primal and dual agree to within numerical errors and show good convergence as the resolution is A.1. Discretization. Here we describe in detail the discretization employed for the primal program on the Swiss cross in the L frame, a program we discussed in Sect. 4.1. The numerical solution requires discretization of the region identified as a fundamental domain in Fig. 10. We explain the discretization using Fig. 38. The square region 0 ≤ x, y ≤ 1/2 has N 2 plaquettes, with N some integer. Each plaquette is a little square with edge given by We introduce a (rational) number k such that k N ∈ Z is the number of plaquettes that fit horizontally (or vertically). Since the distance from the origin to the vertical edge is s /2 and the edge of a plaquette is we have While the function φ 1 lives on the lattice points, its derivatives and the metric are defined to live at the center of the plaquettes. We label the plaquette by the values (i, j) of the lattice point on the lower left of the plaquette. Thus derivatives, for example, are given by Note that on account of (4.12) relating φ 1 (= φ) and φ 2 derivatives we have The value of the scale factor at the center of the (i, j) plaquette is called [i, j] and, as indicated in the program (4.18), we have two inequalities to consider: Note that by defining  We need also the Gaussian curvature K of our metric ds 2 = (dx 2 + dy 2 ). The value is In a planar discretization the Laplacian of a function is computed using 5 points, one at the center where we evaluate the Laplacian. There are two natural options, a cross  . 39. a and b Extremal area A on the s = 2 Swiss cross obtained numerically from the primal (blue) and dual (orange) programs, versus the linear resolution "res" of the lattice discretization. c Absolute value of difference between the optimal A at a given resolution and the optimum value A * in (A.15) (the average of the primal and dual results at the highest resolution). The dotted black line is (res) −3 , showing that convergence is in excellent agreement with a power law with an exponent around −3 configuration and a square configuration. For a function f (x, y) the Laplacian at the origin using these configurations is (A.14) A.2. Convergence. In this "Appendix" we present data showing that the discretized primal and dual programs agree with each other and converge as the lattice resolution is increased. For concreteness, we focus on the s = 2 Swiss cross, in the L frame. Very similar convergence behavior was observed for the punctured torus. We start by studying the convergence of the extremal area as the resolution is increased. We define the linear resolution as the number of plaquettes on the long edge of the fundamental domain. In the notation of subsection A.1 the linear resolution is 2N . The total number of plaquettes in the fundamental domain is 3N 2 . Figure 39 shows the extremal area computed from the primal and dual programs at linear resolutions ranging from 4 to 128 plaquettes. The highest-resolution value for the extremal area is obtained from either the primal or dual, which give the same result to seven significant figures: A * = 2.806 970.
(A. 15) The error will be estimated below as ±10 −6 , or ±1 in the last digit. There are two notable features in the numerical results. The first one is that the optimal values of the primal and the dual track each other quite closely as the resolution changes, with the primal value a tiny bit higher than the dual. At low resolution, both underestimate the extremal area. It might have been expected that the primal program, since it is a minimization program, would overestimate the area. However, while the primal objective evaluated exactly on any continuum configuration bounds the true minimum, in our numerical implementation the configuration is defined on a lattice and the objective involves discretized derivatives. Therefore there is no inconsistency in having a value of the objective which lies below the true continuum minimum. While we do not know why the primal and dual optimal values track each other so closely as the resolution is increased, one plausible explanation is that the discretized programs, at the same resolution, are actually related to each other (perhaps only approximately) by strong duality.
The second notable result is that the primal and dual optimal values both converge to a common value at a rate consistent with a power law with exponent −3. Indeed, defining the error A as a function of the linear resolution "res" (equal to 2N in the language of "Appendix A. It would be interesting to try to derive this convergence rate from a first-principles analysis of the discretized programs. We have not attempted such an analysis; instead we merely take the agreement between the primal and dual and their joint rapid convergence as indications that the errors introduced by the discretization are under control and the results are reliable. In this spirit we estimate the error in (A.15) as ±10 −6 , calculated by assuming that the error depends on the resolution as in (A.17). A more conservative error estimate is to take the difference between the highest and second-highest resolutions (res = 128 and res = 64), about 5 × 10 −6 . Next, we study the extremal metric, or more precisely its line element ρ = √ . In Fig. 40, we show the convergence of ρ for both the primal and dual at a representative point, the midpoint between the bottom-left corner and the inner corner of the fundamental domain (corresponding to x = y = 1 4 in "Appendix A.1"). Just as they did for the area, the metrics obtained from the primal and dual track each other closely and converge as a function of resolution like a power law with an exponent of roughly −3. At the highest resolution, the value of ρ at the representative point differs between the primal and dual by less than 1 part in 10 6 . The difference between the primal and dual is not uniform over the fundamental domain, being largest near the inner corner, but is everywhere less than 1 part in 10 3 . Finally, in Fig. 41, we show the convergence of ln ρ in the dual program over the whole fundamental domain as the resolution is increased.