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A diameter bound for compact surfaces and the Plateau-Douglas problem

In this paper we give a geometric argument for bounding the diameter of a connected compact surface (with boundary) of arbitrary codimension in Euclidean space in terms of Topping's diameter bound for closed surfaces (without boundary). The obtained estimate is expected to be optimal for minimal surfaces in the sense that optimality follows if the Topping conjecture holds true. Our result directly implies an explicit nonexistence criterion in the classical Plateau-Douglas problem. We exhibit examples of boundary contours to ensure that our criterion is of different type from classical criteria based on the maximum principle and White's criterion based on a density estimate.


Introduction
In this paper let M ֒→ R n denote an immersed compact surface (two-dimensional if not specified) in Euclidean n-space, n ≥ 3. In 1998 [22] Topping poses the conjecture that for n = 3, if M is a connected closed surface (without boundary), then where d(M ) := max p,q∈M֒→R n |p − q| denotes the extrinsic diameter and H denotes the mean curvature vector with the convention that |H| ≡ 1 holds for a unit sphere.
In particular, if we define the constant C T (n) := inf 1 d(M ) M |H| M ֒→ R n connected closed surface , the Topping conjecture predicts that C T (3) = π. In this problem an optimal shape is expected to be a long cylinder with capped ends, which directly implies that C T (n) ≤ π. The exact value of C T (n) is still open, but it is currently known that C T (n) ≥ π/16 (n = 3, 4), π/32 (n ≥ 5). (1.2) At this time it is not clear even whether or not the value of C T (n) depends on n.
In this paper we obtain a similar type of diameter bound for general compact surfaces (with boundary) by directly relating with the constant C T (n). Theorem 1.1. Let M ֒→ R n be an immersed connected compact surface, where n ≥ 3. Then where ℓ(∂M ) denotes the length of the boundary ∂M in R n .
In particular, if M is a minimal surface (H ≡ 0), then Theorem 1.1 implies that This directly gives a nonexistence criterion in the classical Plateau-Douglas problem; see Section 3 for details. In addition, it would be worth indicating that the prefactor in (1.4) becomes optimal if the Topping conjecture holds true. Indeed, if C T (n) = π, then (1.4) leads to ℓ(∂M )/d(M ) ≥ 2, while this 2 cannot be improved due to a planar surface spanning a closed curve consisting of two parallel long segments capped by semicircles. The above observation naturally raises a codimension-extension problem involving diameter. More precisely, we expect that the following strict inequality holds for every connected compact minimal surface M in R n with n = 3 or higher for which C T (n) = π holds. Since the inequality is obviously true for n = 2 (where M is just flat), this problem asks an extendibility with respect to codimension. Our expectation seems reasonable because this kind of codimension-extension problem is extensively studied for the isoperimetric inequality and now verified up to codimension two by Brendle; see [2] and references therein. Now we review what is known of the Topping conjecture. The lower bound C T (n) ≥ π/32 in (1.2) is first proved by Topping himself [23], where higher dimensional M and the intrinsic diameter are treated. The improved lower bound C T (n) ≥ π/16 for n ≤ 4 follows because Topping's argument is based on the Michael-Simon Sobolev inequality, for which a sharp constant is obtained up to codimension 2 by Brendle [2]; see also Appendix A. The exact form of the Topping conjecture (1.1) is quite delicate; positive answers are known only for convex surfaces (classically), constant mean curvature surfaces [21], and simply-connected axisymmetric surfaces [12]. Our present study confirms that the Topping conjecture is not inconsistent with the aforementioned codimension-extension problem.
Concerning surfaces with boundary, Menne-Scharrer's recent result [11] provides a diameter bound in a fairly general framework of varifolds, which is new even for smooth surfaces: For 1 < m < n and a connected compact m-dimensional submanifold M m of R n , the intrinsic diameter d int (M ) has a bound of the form If m = 2, the last term is interpreted as the length ℓ(∂M ). This result extends both Topping's result for closed submanifolds [23] and also Paeng's result for geodesically convex compact surfaces (m = 2) [18]. The constant c m is chosen in a somewhat involved way. Compared to Menne-Scharrer's result, our present study is less general e.g. in view of dimension m, but our focus is optimality rather than generality. Our main contribution is giving not only an explicit constant but also a direct relation with the case of closed surfaces by a completely different simple proof.
In Section 2 we prove Theorem 1.1. The proof is based on the simple idea to enclose a given compact surface by a thin closed surface; the compact surface is doubled and glued along the boundary. Our method crucially relies on twodimensionality of surfaces. In Section 3 we indicate that Theorem 1.1 (or Menne-Scharrer's preceding result) directly implies a novel type of nonexistence criterion in the Plateau-Douglas problem by exhibiting several concrete examples. In Appendix A the lower bound in (1.2) for n ≤ 4 is verified for the reader's convenience. out that Brendle's recent result implies an improved diameter bound. He also thanks Genki Hosono and Yuichi Ike for stimulating discussions. This work is supported by JSPS KAKENHI Grant Number 18H03670 and 20K14341, and by Grant for Basic Science Research Projects from The Sumitomo Foundation.

Bounding diameter for surfaces with boundary
In this section we prove Theorem 1.1. The key step is to ensure the following Proposition 2.1 (Construction of closed surfaces). For any connected compact surface M immersed into R n with ∂M = ∅, there exists a sequence {Σ k } k of connected closed surfaces in R n such that We first observe that Theorem 1.1 immediately follows from Proposition 2.1.
Proof of Theorem 1.1. Let M be given. We may suppose that ∂M = ∅. Take a sequence {Σ k } in Proposition 2.1. Then by definition of C T (n) we have Taking the limit that k → ∞, we deduce (1.3) from (2.1) and (2.2).
In the remaining part we prove Proposition 2.1. The primal idea is to enclose a compact surface M by a thin closed surface Σ k . In the simplest case of codimension one (n = 3) and M embedded, we may just take Σ k as the boundary of the 1 k -neighborhood of M in R 3 for k ≫ 1 as in Figure  1. Then, roughly speaking, the enclosing surface consists of two parts; a "nearly double-cover" of M and "curved half cylinders of radius 1/k" along the boundary ∂M . The curvature energy of the former part clearly converges to twice that of M as k → ∞, while the latter yields π 2 ℓ(∂M ) as a singular limit. The above idea can be extended to non-embedded surfaces almost straightforwardly, but not very directly to higher codimensions since we have no canonical choice of normal directions. In order to give a unified proof for an arbitrary codimension, we slightly modify our idea; we double M and glue their boundaries via a new surface made by a teardrop-shaped curve (with a cusp) as in Figure 2, so that the resulting closed surface looks "pressed" to touch M from both sides. This modification allows us to only think of the boundary construction, which is essentially reduced to the case of codimension one by using a three-dimensional frame associated with each boundary curve. Before entering the proof we prepare a teardrop curve for later use.
where κ and s denote the curvature and the arclength parameter, respectively.
Proof. By approximation we only need to construct γ k of class C 1,1 (= W 2,∞ ) and piecewise smooth. Define γ k by connecting the origin and the half-circle 1] is any fixed smooth function such that f (0) ≡ 0 around 0 and f (1) ≡ 1 around 1. By construction all the assertions are trivial except for (2.5). Convergence (2.5) is also simply confirmed since the total absolute curvature is nothing but the total variation of the tangential angle function; for the half-circle C k |κ|ds = π, while 1 k f → 0 smoothly so that graph(± 1 k f ) |κ|ds → 0. We are now in a position to give a precise proof of Proposition 2.1.
Proof of Proposition 2.1. Fix any connected compact surface M immersed into R n . By approximation it suffices to construct a sequence of surfaces of class C 1,1 (= W 2,∞ ) and piecewise smooth.
Step1: Construction. We first prepare a three-dimensional frame {e 1 , e 2 , e 3 } along the boundary ∂M for a gluing procedure. For each of the boundary components ∂M i , i = 1, . . . , N , being a closed curve of length L i := ℓ(∂M i ) > 0, we let c i : R/L i Z → ∂M i ⊂ R n be a unit-speed parameterization. Now with each ∂M i we assign the (global) vector field e 1 := ∂ s c i , the unit tangent, and e 2 to be the outward pointing unit conormal of the surface M at the boundary. Then we choose one more vector field e 3 orthogonal to both e 1 and e 2 . Note that such an e 3 always exists. Indeed, for n = 3 we may just take e 3 = e 1 × e 2 . For n ≥ 4, if we let N ′ denote the subbundle of the normal bundle to ∂M i in R n defined by N ′ := span{e 2 } ⊥ , where ⊥ stands for the orthogonal complement, then N ′ is locally trivialized as I × R n−2 , where I is an interval, and hence there is no (topological) obstruction to choose a global section of N ′ as n − 2 ≥ 2. Slightly abusing the notation, we also mean by e j just a smooth map from R/L i Z to S n−1 ⊂ R n .
We then define a surface S along the boundary ∂M , which will be used to close two copies of M . Given ε > 0, integer k > 0, and i = 1, . . . , N , using the teardrop curve ( Note that since {e 1 , e 2 , e 3 } are smooth maps on a compact set, there is someε > 0 such that for any ε ∈ (0,ε) the patch S is regular, i.e., ∂ s S and ∂ σ S are linearly independent; the smallness depends only on the frame but not on k thanks to (2.4). Now we define a piecewise smooth closed surface Σ ε,k by gluing the boundaries of two copies of M , say M and M ′ , via S. More precisely, for all i we glue ∂M i to one boundary component S(R/L i Z×{0}), and ∂M ′ i to the other S(R/L i Z×{L ε }), keeping their shapes. Notice that Σ ε,k is of class C 1,1 thanks to boundary condition (2.3). In addition, Σ ε,k = M ∪ M ′ ∪ S is connected since S is joined with both M and M ′ so that M and M ′ are also joined via S.
Step 2: Quantitative behavior. In what follows we will define Σ k := Σ ε k ,k for a well-chosen ε k to satisfy the desired properties, (2.1) and (2.2).
Concerning diameter, by our construction, clearly d(M ) ≤ d(Σ ε,k ) holds. In addition, thanks to (2.4), for any p ∈ Σ ε,k there is p ′ ∈ M such that |p − p ′ | ≤ 2ε, and hence max p,q∈Σ ε,k |p − q| ≤ max p ′ ,q ′ ∈M |p ′ − q ′ | + 4ε. We thus find that Concerning the curvature energy, by definition we have We now confirm that for each i and k, To this end we explicitly represent the integral. Let g = (g jl ) 1≤j,l≤2 denote the pull-back metric of the Euclidean metric of R n under S in the local coordinate (s, σ); namely, g jl = ∂ j S · ∂ l S where ∂ 1 := ∂ s and ∂ 2 := ∂ σ . Recall that for S the induced area measure is given by det (g jl )dsdσ, while the second fundamental form A by A jl := (∂ jl S) ⊥ , where ⊥ denotes the normal projection, and the mean curvature vector H by H = 1 2 g jl A jl (up to the sign, through the Einstein notation), where (g jl ) := (g jl ) −1 . Thus we have Now we expand the integrand in terms of ε by explicit calculations. Since and since e 1 , e 2 , e 3 are orthogonal, we have as ε → 0, ; here and in the sequel asymptotic notations are independent of the parameters (s, σ). These imply that Then, noting that and that |g 11 , we now only need to compute the term , we compute e ⊥ j for j = 2, 3. By using the representation , and noting that the last term is of lower order, we deduce that . Then a direct computation shows that and hence |g 22 A 11 | = ε|κ| + o(ε). In summary, we have |g jl A jl | det (g jl ) = |κ| + o(1).

Nonexistence theorem in the Plateau-Douglas problem
In this section we discuss the Plateau-Douglas problem, also known as the Douglas problem or simply the (general) Plateau problem, focusing on the simplest case of codimension one. The Plateau-Douglas problem asks whether a connected minimal surface M ֒→ R 3 spans a given boundary contour Γ ⊂ R 3 that consists of mutually disjoint smooth Jordan curves.
In this problem the existence of a connected solution sensitively depends on the geometry of Γ, in contrast to the case that Γ is connected (the classical Plateau problem). A simple phenomenological example is a catenoid-shaped soap film spanning two parallel circular wires; such a connected film exists if the wires are close to each other, but the film will pinch off if the wires are pulled apart. Up to rescaling, it is equivalent to shrink the circles while keeping the distance, as in Figure 3. (We remark that Figure 3 describes only stable catenoids; however, at any rate, below a critical radius the boundary admits no catenoid regardless of the stability.) Up to now some sufficient conditions on Γ for existence of connected minimal surfaces are known, including the celebrated Douglas condition (see e.g. [4,Section 8]), but they do not completely characterize the existence. By this reason there are also many studies to explore nontrivial necessary conditions. Our result (1.4) combined with (1.2) and the fact that d(∂M ) ≤ d(M ) directly implies a nonexistence criterion, which also explains certain pinching phenomena: Then there exists no connected minimal surface spanning Γ. In the rest of this section we observe that this kind of criterion is not covered by existing criteria, by exhibiting some concrete examples.
After Nitsche's foundational studies, many authors establish several kinds of nonexistence criteria (see e.g. [13,14,15,1,16,8,17,3,19,9,6,10] and also [5,Chapter 4]), most of which are based on the maximum principle (except for some results using special geometry). Roughly speaking, such results assert that no connected solution spans a boundary being "divided into two parts far from each other". For example, a well-known criterion of cone type, which is first obtained by Hildebrandt [8] and later improved by Osserman-Schiffer [17], ensures that no solution exists if Γ is the union of Γ 1 and Γ 2 (not necessarily connected) such that where τ is a unique positive solution to cosh τ = τ sinh τ . The prefactor sinh 2 τ ≈ 2.27 is known to be optimal, cf. [17]. This kind of criterion is sufficient for explaining the pinching of a catenoid. However, it does not necessarily cover the case where many circles are simultaneously pulled apart in some random directions, or equivalently, shrunk into several randomly distributed points, cf. Figure 4. Recently, White gives an essentially different criterion [24,Corollary 9] (see also [20]) by making use of an extended monotonicity theorem for density [7]. It asserts that no solution spans Γ if there is a decomposition Γ = Γ 1 ∪ Γ 2 such that This looks similar to our one (3.1) as both the conditions require smallness of length, and in fact shares the common feature of being "less unidirectional". We shall discuss "multi-directionality" through a concrete example of a boundary contour that is covered by both (3.1) and (3.3) but not by (3.2). Example 3.3 (Small circles on the sphere: Fixed centers). Let X be any finite set in the unit sphere S 2 ⊂ R 3 containing more than one point. Let r X be the packing radius of X in S 2 , i.e., half of the minimal geodesic distance of two distinct points in X ⊂ S 2 , and also R X the covering radius, i.e., the minimal R > 0 such that the R-neighborhood of X in S 2 covers the whole of the sphere S 2 . Let Γ ε be the union of all the (mutually disjoint) geodesic circles centered at the points in X of radius ε < r X . Then our condition (1.1) is satisfied by Γ ε for any small ε, since d(Γ ε ) → d(X) > 0 while ℓ(Γ ε ) → 0 as ε → 0. Similarly, White's condition (3.3) is also satisfied for any small ε, since if we take Γ ε,1 to be one circle and Γ ε,2 all the remaining circles, then lim ε→0 dist(Γ ε,1 , Γ ε,2 ) ≥ 2rX π . However, the cone condition (3.2) may be violated depending on the choice of X. More precisely, if X is sufficiently dense, say R X ≤ 1 100 , then Γ ε cannot be separated by any cone K ′ congruent to K; indeed, if K ′ separates Γ ε ⊂ S 2 , then the center of K ′ must lie inside S 2 , and this constraint implies a universal lower bound of the (geodesic) inradius of S 2 \ K ′ so that in particular R X > 1 100 must hold. In addition, our condition (3.1) is qualitatively independent from (3.3). To observe this we shall perturb a given Γ by adding very small circles. For such a perturbation, condition (3.1) is generally rigid, but condition (3.3) may be violated depending on the positions since its left-hand side may considerably decrease.
From another point of view we may say that (3.1) is in a sense "more multidirectional" than (3.3), as is seen in the next example, which is similar to Example 3.3 but allows more densely distributed circles.
Example 3.4 (Small circles on the sphere: Increasing centers). Let X ε be an ε-net of S 2 with ε ≪ 1; namely, if we let r Xε (resp. R Xε ) denote the packing (resp. covering) radius as in Example 3.3, then ε/2 ≤ r Xε < R Xε ≤ ε. Let Γ ε be the union of all the geodesic circles centered at the points in X ε of radius ε 5 2 (≪ ε). Note that the cardinality of X ε is of the form |X ε | ∼ ε −2 , and hence ℓ(Γ ε ) ∼ ε 5 2 · ε −2 = ε 1 2 . Then (3.1) is still satisfied since d(Γ ε ) → 1 and ℓ(Γ ε ) → 0 as ε → 0. However, (3.3) is violated since dist(Γ ε,1 , Γ ε,2 ) ε holds for an arbitrary choice of decomposition Γ ε,1 and Γ ε,2 , so that dist(Γ ε,1 , We finally remark that the above multi-directional criteria (3.3) and (3.1) require smallness of ℓ(Γ), although unidirectional results such as (3.2) need not require any smallness of first order. To find a (qualitative) common roof, it would be a natural direction to explore a "zeroth order" multi-directional criterion. For example: Given an arbitrary finite set X ⊂ R 3 containing more than one point, can one find ε = ε(X) > 0 such that if the boundary ∂M of a connected compact minimal surface M is contained in the ε-neighborhood U ε (X) of X, then there is a connected component of U ε (X) that does not intersect with ∂M ?
Notice that such an ε clearly exists if X is separated by the (open) cone K in (3.2) up to a rigid motion, since so is U ε (X) for any small ε in that case. Notice also that such an inclusion property always holds if X has a sufficiently small number of points. However, it seems still open for X being more uniformly distributed. A next step would be to ask which quantity of X the choice of ε depends on.
Appendix A. Lower bound of Topping's constant In this appendix we obtain the lower bound π/16 in (1.2) for codimension ≤ 2. In fact we only need to follow Topping's original argument [23] only with two very minor changes: One is about the constant σ in the Michael-Simon Sobolev inequality (A.1), for which we use Brendle's sharp constant σ = 2 √ π (instead of σ = √ 2π used in [23]): The other is that, accordingly, in Lemma A.1 below we may choose δ = π/4 (instead of δ = π/8 used in [23]). The restriction of dimension n ≤ 4 is because Brendle's result is not sharp for codimension larger than 2.
We first recall the Michael-Simon Sobolev inequality with the sharp constant obtained in [2]. Here we focus on a closed surface M in R 4 for our purpose. In this case we have for every non-negative smooth function f on M , and hence also for Lipschitz f by a standard approximation argument. (Note that the prefactor 2 of Hf L 1 (M) comes from the averaging factor of H.) The sharp inequality (A.1) then implies the following lemma, which is a sharpened version of [23, Lemma 1.2]. For completion we give a proof, which is just a straightforward modification of Topping's argument [23] but in fact possesses a somewhat delicate issue; see Remark A.2 below for details. Given µ > 0, we define a Lipschitz cut-off function f on M by f ≡ 1 inside B(q, r), f ≡ 0 outside B(p, r + µ), and f := 1 − 1 µ (dist M (·, p) − r) in the annulus B(p, r + µ) \ B(p, r). Applying (A.1) to this f , and dropping p in V (p, R) for simplicity, we have σV (r) 1 2 ≤ σ f L 2 (M) ≤ 1 µ (V (r + µ) − V (µ)) + 2 H L 1 (B(p,r+µ)) .
Recall that V is locally Lipschitz, where the Lipschitz constant is bounded in terms of the Ricci curvature. Hence, letting µ ↓ 0, we deduce that for a.e. r, σV 1 2 ≤ dV dr + 2 H L 1 (B(p,r)) .
Remark A.2. A core of Topping's argument is obtaining a positive lower bound for U := V /r 2 by using the structure of differential inequality (A.3). As long as we follow this strategy, the best possible choice of δ would be π/4. Indeed, (A.3) is equivalent that rU ′ + 2( √ U − π/4) 2 ≥ 2(π/4 − δ); the right-hand side is negative if δ > π/4, and hence it is not ruled out that U (r) → −∞ as r → ∞ (e.g. in the case of equality).
Once we get Lemma A.1, by the completely same covering argument as in [23,Section 3] we deduce that Since δ = π/4 here, we now obtain the desired lower bound π/16 in (1.2).