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An Introduction to Potential Theory in Calibrated Geometry

We introduce and study the notion of plurisubharmonic functions in calibrated geometry. These functions generalize the classical plurisubharmonic functions from complex geometry and enjoy their important properties. Moreover, they exist in abundance whereas the corresponding pluriharmonics are generally quite scarce. A number of the results established in complex analysis via plurisubharmonic functions are extended to calibrated manifolds. In particular, the notion of pseudo-convexity for a calibrated manifold (X,\phi) is introduced and studied. Analogues of totally real submanifolds are also introduced and used to construct enormous families of strictly \phi-convex spaces with every topological type allowed by Morse Theory. Specific calibrations are used as examples throughout.


Appendices
Appendix A: Submanifolds which are φ-Critical.
Appendix B: Constructing φ-Plurisubharmonic Functions. Appendix C: Structure of the Core.

Introduction.
Calibrated geometries, as introduced in [HL 1 ], are geometries of distinguished submanifolds determined by a fixed, closed differential form φ on a riemannian manifold X. The basic example is that of a Kähler manifold (or more generally a symplectic manifold, with compatible almost complex structure) where the distinguished submanifolds are the holomorphic curves. However, there exist many other interesting geometries, each carrying a wealth of φ-submanifolds, particularly on spaces with special holonomy. The relationship between spinors and calibrations established in [DH] provides additional interest. Calibrated manifolds have attracted particular attention in recent years due to their appearance in generalized Donaldson theories ( [DT], [Ti]) and in modern versions of string theory in Physics ( [GLW], [GP], [AFS], [Her], [MS], [G], [EM], [GW], [MC] for example).
Unfortunately, analysis on these spaces (X, φ) has been difficult, in part because there is generally no reasonable analogue of the holomorphic functions and transformations which exist in the Kähler case. However, in complex analysis there are many important results which can be established using only the plurisubharmonic functions (cf. [Ho], [D]). It turns out that analogues of these functions exist in abundance on any calibrated manifold, and they enjoy almost all the pleasant properties of their cousins from complex analysis. The point of this paper is to introduce and study these functions and related notions of convexity.
In a sequel [HL 2 ] these notions will be related to φ-positive currents and their boundaries, generalized Jensen measures, and other geometric objects on a calibrated manifold.
We begin by defining our notion of φ-plurisubharmonicity for smooth functions on any calibrated manifold (X, φ). In the Kähler case they are exactly the classical plurisubharmonic functions. We then study the basic properties of these functions, and subsequently use them to establish a series of results in geometry and analysis on (X, φ).
A fundamental result is that: The restriction of a φ-plurisubharmonic function to a φ-submanifold M is subharmonic in the induced metric on M .
Any convex function on the riemannnian manifold X is φ-plurisubharmonic. Moreover, at least locally, there exists an abundance of φ-plurisubharmonic functions which are not convex.
The definition of φ-plurisubharmonicity extends from smooth functions to arbitrary distributions on X. Such distributions enjoy all the nice properties of generalized subharmonic functions. In this paper, however, we shall focus mainly on the smooth case, except for Section 3.
To define φ-plurisubharmonic functions on a calibrated manifold (X, φ) where deg(φ) = p, we introduce a second order differential operator H φ : C ∞ (X) → E p (X), the φ-Hessian, given by where Hessf is the riemannian hessian of f and λ φ : End(T X) → Λ p T * X is the bundle map given by λ φ (A) = D A * (φ) where D A * : Λ p T * X −→ Λ p T * X is the natural extension of A * : T * X → T * X as a derivation.
When the calibration φ is parallel there is a natural factorization where d is the de Rham differential and d φ : C ∞ (X) → E p−1 (X) is given by In general these operators are related by the equation: Recall that a calibration φ of degree p is a closed p-form with the property that φ(ξ) ≤ 1 for all unit simple tangent p-vectors ξ on X. Those ξ for which φ(ξ) = 1 are called φ-planes, and the set of φ-planes is denoted by G(φ). With this understood, a function f ∈ C ∞ (X) is defined to be φ-plurisubharmonic if H φ (f )(ξ) ≥ 0 for all ξ ∈ G(φ). It is strictly φ-plurisubharmonic at a point x ∈ X if H φ (f )(ξ) > 0 for all φ-planes ξ at x. In a similar fashion, f is called φ-pluriharmonic if H φ (f )(ξ) = 0 for all ξ ∈ G(φ). Denote by PSH(X, φ) the convex cone of φ-plurisubharmonic functions on X.
When X is a complex manifold with a Kähler form ω, one easily computes that d ω = d c , the conjugate differential. In this case, H ω = dd ω = dd c and the ω-planes correspond to the complex lines in T X. Hence, the definitions above coincide with the classical notions of plurisubharmonic and pluriharmonic functions on X.
With this said, we must remark that in many calibrated manifolds the φ-pluriharmonic functions are scarce. For the calibrations on manifolds with strict G 2 or Spin 7 holonomy, for example, every pluriharmonic function is constant. For the Special Lagrangian calibration φ = Re{dz}, every φ-pluriharmonic function f defined locally in C n is of the form f = a+q where a is affine and q is a traceless Hermitian quadratic form (cf. [Fu] and Proposition 1.10.) Nevertheless, as we stated above, the φ-plurisubharmonic functions in any calibrated geometry are locally abundant.

ÄÄÁÈÌÁ ÄÁ Ê ÌÁÇAEËº
This brief section is an introduction to the theory of φ-plurisubharmonic distributions. A very mild condition on the calibration is needed to ensure "ellipticity", namely, G(φ) should involve all the variables. (This is stronger than requiring that the calibration φ involve all the variables. See Example 2 in Section 3.) Under this assumption, each φ-plurisubharmonic distribution is, in fact, L 1 loc (locally Lebesgue integrable) and has a canonical point-wise representative which is [−∞, ∞)-valued and upper semi-continuous, given by the limit of the means over balls. The usual properties of plurisubharmonic function in complex analysis are valid for these φ-plurisubharmonic functions. See [HL 4 ] for a more comprehensive development, which is also calibration independent.
Beginning with Section 4 the φ-plurisubharmonic functions are used to study geometry and analysis on calibrated manifolds. The first concept to be addressed is the analogue of pseudoconvexity in complex geometry. ÈÊÇÈÇËÁÌÁÇAE 5.9. Let Ω ⊂⊂ X be a compact domain with smooth boundary. Suppose φ is parallel and consider the function δ = dist (•, ∂Ω). If −logδ is φ-plurisubharmonic near ∂Ω, then ∂Ω is φ-convex.
We note that boundary convexity can be interpreted geometrically as follows. Let II denote the second fundamental form of the hypersurface ∂Ω oriented by the outwardpointing normal. Then ∂Ω is φ-convex if and only if trace(II ξ ) ≤ 0 for all φ-planes ξ which are tangent to ∂Ω. In the strict case one also has the following.
ÌÀ ÇÊ Å 5.14. Let (X, φ) be a strictly φ-convex manifold and Ω ⊂⊂ X a domain with smooth boundary. Then the following are equivalent.
(iii) There exists a smooth defining function ρ for ∂Ω which is strictly φ-plurisubharmonic on a neighborhood of Ω.
φ¹ Ê ËÍ Å AEÁ ÇÄ Ë AE ËÌÊÁ ÌÄ φ¹ ÇAEÎ ËÍ ÇÅ ÁAEËº We next examine the analogues in calibrated geometry of the totally real submanifolds in complex analysis. Using the methods of [HW 1,2 ] we then show how to construct strictly φ-convex manifolds in enormous families with every topological type allowed by Morse theory.
Let (X, φ) be any fixed calibrated manifold. A closed submanifold M ⊂ X is called φ-free if there are no φ-planes tangential to M , i.e., no ξ ∈ G(φ) with span ξ ⊂ T M . Note that M is automatically φ-free if it is φ-isotropic, that is, if φ M ≡ 0. . Any submanifold of dimension < p is φ-free, and generic local submanifolds of dimension p are φ-free. Furthermore, any submanifold of a φ-free submanifold is again φ-free.
The φ-free dimension of (X, φ), denoted fd(φ), is defined to be the largest dimension of a φ-free vector subspace of T x X for x ∈ X. The first result is the following generalization of the Andreotti-Frenkel Theorem [AF] for Stein manifolds. ÌÀ ÇÊ Å 6.2. Suppose (X, φ) is a strictly φ-convex manifold. Then X has the homotopy type of a CW complex of dimension ≤ fd(φ).
For a Kähler manifold (X, ω) of complex dimension n, the ω-free dimension is n and the ω-free submanifolds are those which are totally real (e.g., the Lagrangian submanifolds). Furthermore the ω p /p!-free dimension is n + p − 1 and a submanifold M is ω p /p!-free if there are no complex p-planes tangent to M at any point.
If (X, φ) is a 7-dimensional G 2 -manifold with associative calibration φ, then fd(φ) = 4. So if X is φ-convex , it has homotopy dimension ≤ 4. Recently, I.Unal [U] has shown that for every connected manifold M of dimension < 4 (compact or non-compact) there exists a strictly φ-convex G 2 -manifold which is homotopy-equivalent to M . The relationship between φ-free submanifolds and convexity is expressed in the next two results.
ÌÀ ÇÊ Å 6.4. Suppose M is a closed submanifold of (X, φ) and let dist 2 M (x) denote the square of the distance to M . Then M is φ-free if and only if the function dist 2 M (x) is strictly φ-plurisubharmonic at each point in M (and hence in a neighborhood of M ).
The existence of φ-free submanifolds ensures the existence of many strictly φ-convex domains in (X, φ). ÌÀ ÇÊ Å 6.6. Suppose M is a φ-free submanifold of (X, φ). Then there exists a fundamental system F (M ) of strictly φ-convex neighborhoods of M , each of which admits a deformation retraction onto M .
This result provides rich families of strictly convex domains. The neighborhoods in F (M ) include the sets {x : dist M (x) < ǫ(x)} for positive functions ǫ ∈ C ∞ (M ) which die arbitrarily rapidly at infinity. As noted, any submanifold of dimension < p is φ-free. Furthermore, any submanifold of a φ-free submanifold is again φ-free.
For example if X is a Calabi-Yau manifold with Special Lagrangian calibration φ, then any symplectic submanifold Y ⊂ X is φ-free, as is any smooth submanifold A ⊂ Y . The topological type of such manifolds A can be quite complicated.
This construction can be refined even further by replacing the submanifold A ⊂ Y with an arbitrary closed subset. It turns out that the following two classes of subsets: (1) Closed subsets A of φ-free submanifolds (2) Zero sets of non-negative strictly φ-plurisubharmonic functions f are essentially the same.
We mention that the operator d φ has been independently found by M. Verbitsky [V] who studied the generalized Kähler theory (in the sense of Chern) on G 2 -manifolds. The authors would like to thank Robert Bryant for useful comments and conversations related to this paper.

Plurisubharmonic Functions
Suppose φ is a calibration on a riemannian manifold X. The φ-Grassmannian, denoted G(φ), consists of the unit simple p-vectors ξ with φ(ξ) = 1, i.e., the φ-planes. An oriented submanifold M is a φ-submanifold, or is calibrated by φ, if the oriented unit tangent space − → T x M lies in G x (φ) for each x ∈ M , or equivalently, if φ restricts to M to be the volume form on M . Let n = dimX and p = degree(φ).
for all smooth functions f on X. Hence where E p (X) denotes the space of C ∞ p-forms on X. This dd φ operator provides a way of defining plurisubharmonic functions in calibrated geometry when the calibration φ is parallel.
If ω is a Kähler form on a complex manifold, then d ω = d c = −J • d is the conjugate differential. Thus, the dd φ -operator generalizes the dd c -operator in complex geometry. Although no analogue of a holomorphic function exists on a calibrated manifold, there is an analogue of the real part of a holomorphic function.
The set of such functions will be denoted PSH(X, φ).
Note that f is φ-pluriharmonic if and only if both f and −f are φ-plurisubharmonic, and that f is partially φ-pluriharmonic if and only if f is φ-plurisubharmonic but not strict at any point.
This modified dd φ -operator is discussed in detail in Section 2. Note that the difference ∇ ∇f φ is a first order operator.
The next result justifies the use of the word plurisubharmonic in the context of a φ-geometry. A calibration φ is integrable if for each point x ∈ X and each ξ ∈ G then the restriction of f to any φ-submanifold M ⊂ X is subharmonic in the induced metric. If φ is integrable, then the converse holds.
Theorem 1.4 is an immediate consequence of the formula This formula follows from the three equations (2.7), (2.12), and (2.15), proved below, and the fact that φ-submanifolds are minimal submanifolds. We continue for the moment to present results whose proofs will be given in Section 2. The φ-plurisubharmonic functions enjoy many of the useful properties of their classical cousins in complex analysis. The next result is useful, in particular if one wishes to only consider smooth φ-plurisubharmonic functions Proof. See Appendix B.
For now we mention a "fundamental" example.
Ê Å ÊÃ (The Abundance of φ-plurisubharmonic Functions). We shall see in the next section that any convex function on the riemannian manifold X is automatically φ-plurisubharmonic. However, there always exist huge families of locally defined φ-plurisubharmonic functions which are not convex. This follows, for example, from duality considerations as in Remark 2.9 below. However, in section 6 we give a general construction of φ-plurisubharmonic functions from any φ-free submanifold, which shows that such functions exist in abundance.

Pluriharmonic Functions
The φ-pluriharmonic functions are a natural replacement for the holomorphic functions in complex geometry. However, while φ-plurisubharmonic functions are abundant, the φ-pluriharmonic functions are often quite scarce. To illustrate this phenomenon we shall sketch some of the basic facts in the "classical" cases.
To begin we note that for some calibrations φ, one has that: while for others this is not true. It is the right hand side that defines pluriharmonicity. If (1.4) holds and the basic map λ φ , defined in section 2, is everywhere injective (as in Example 1.14), then the only pluriharmonic functions are the affine functions, i.e., the functions with parallel gradient. Note that if f is affine, then ∇f splits the manifold locally as a riemannian product X = R × X 0 . ÅÈÄ 1.8. (Complex geometry). Let ω be a Kähler form on a complex manifold X. Then d ω = d c is the conjugate differential, dd c f is the complex hermitian Hessian of f , G(ω) is the grassmannian of complex lines, and the statement (1.4) is valid. In particular, the ω-pluriharmonic functions are just the classical pluriharmonic functions on X.
For the higher divided powers Ω p = 1 p! ω p one computes that dd Ω p f = Ω p−1 dd c f . Furthermore, it can be deduced from the discussion in Remark 2.13 that (1.4) holds in this case. Therefore, the Ω p -pluriharmonic functions are also just the classical pluriharmonic functions. ÅÈÄ 1.9. (Quaternionic-Kähler geometry). Let H denote the quaternions and consider H n as a right-H vector space. Each of the complex structures I, J, K (right multiplication by i, j, k) determines a Kähler form ω I , ω J , ω K respectively. The 4-form on H n ≡ R 4n is a calibration with G(Ψ) consisting of the oriented quaternion lines in H n . In this case, dd Ψ f ≡ 0 if and only if Hessf ≡ 0. However, the assertion (1.4) is not valid in this case, and in fact there is a rich family of Ψ-pluriharmonic functions. For example, if f is ω I -pluriharmonic, then f is Ψ-pluriharmonic. Hence, so is any ω-pluriharmonic f where ω = aω I + bω J + cω K with a 2 + b 2 + c 2 = 1. It is well known that the only Ψ-submanifolds in H n are the affine quaternion lines. Of course the calibration (1.5) exists on any quaternionic Kähler manifold, i.e., one with Sp n · Sp 1 -holonomy. (See [GL] for examples.) With this full holonomy group it seems unlikely that there are many Ψ-pluriharmonic functions. However, if the holonomy is contained in Sp n , they exist in abundance as seen in the next example. ÅÈÄ 1.10. (Hyper-Kähler manifolds). Let (X, ω I , ω J , ω K ) be a hyper-Kähler manifold. Then X carries several parallel calibrations. There are, of course, the Kähler forms ω = aω I + bω J + cω K with a 2 + b 2 + c 2 = 1, and two others of particular interest.
(2) Consider the generalized Cayley form Ξ ≡ 1 2 (ω 2 I − ω 2 J − ω 2 K ). For this calibration there exist no interesting pluriharmonic functions, at least in dimension 8, but there are many Ξ-submanifolds (cf. [BH]). ÅÈÄ 1.11. (Double point geometry). Let φ = dx 1 ∧ · · · ∧ dx n + dy 1 ∧ · · · ∧ dy n in R 2n for n ≥ 3. The only φ-planes are those parallel to the x or y axes. An easy calculation shows that dd φ f = 0 if and only if f (x, y) = g(x) + h(y) for harmonic functions g and h. However, a function f (x, y) is φ-pluriharmonic if and only if it is harmonic in x and y separately. This is a simple example where (1.4) fails.
In all the following examples φ-pluriharmonic functions are quite scarce. Consequently, Lei Fu [Fu] has described all the φ-pluriharmonic functions. ÈÊÇÈÇËÁÌÁÇAE 1.13. Let f be a Special Lagrangian pluriharmonic function defined locally on C n , n ≥ 3. Then f = A + Q where A is affine and Q is a traceless hermitian quadratic function.
Proof. If dd φ f = 0 and n ≥ 3 (so that Z ij and Z ij are of different bi-degrees), then (1.1) implies that ∂ 2 f ∂z i ∂z j = 0 for all i, j. Therefore, all third partial derivatives of f are zero. For polynomials of degree ≤ 2 the result is transparent from (1.1). ÅÈÄ 1.14. (Associative, Coassociative and Cayley geometry). Consider one of the calibrations: where O denotes the octonions. As in the Special Lagrangian case one can show that (1.4) is valid for each of these calibrations. Furthermore, an application of representation theory shows the maps λ φ , λ ψ and λ Φ are injective. These calculations carry over to manifolds with G 2 or Spin 7 -holonomy to establish the following. ÈÊÇÈÇËÁÌÁÇAE 1.15. Let X be a manifold with holonomy contained in G 2 or Spin 7 and having dimension 7 or 8 respectively. Suppose φ is a parallel calibration on X of one of the three types above. Then every φ-pluriharmonic function on X is affine. Moreover, if the holonomy is exactly G 2 or Spin 7 , every φ-pluriharmonic function is constant.
Proof. The first assertion follows because (1.4) is valid and the λ-maps are injective. The second follows because any non-constant affine function on X would reduce its holonomy to a subgroup of {1} × SO n−1 . ÅÈÄ 1.16. (Lie group geometry). Let G be a compact simple Lie group with Lie algebra g, defined as the set of left-invariant vector fields on G.
1) Consider the fundamental 3-form φ on G defined by x, [y, z] and normalized to have comass one. Calculations indicate that in all but a finite number of cases nonconstant pluriharmonic functions do not exist, however there are φ-submanifolds, namely the "minimal" SU 2 -subgroups (cf. [B], [T], [Th]).
2) Consider * φ. The * φ-submanifolds are given by certain components of the cut locus C. Is G − C strictly φ-convex? ÅÈÄ 1.17. (Gromov manifolds). By a Gromov manifold we mean an ensemble (X, ω, J, ·, · ) where (X, ω) is a symplectic manifold, J is an almost complex structure on X and ·, · is a riemannian metric with the property that Every symplectic manifold has many Gromov structures. Generically the almost complex structure J is not integrable, and the only ω-pluriharmonic functions are the constants. However, there are generally many ω-submanifolds (the pseudo-holomorphic curves) and there are many ω-plurisubharmonic functions as we shall see below. It is important to note here that the operator dd c is not appropriate for this context since ∇ω = 0. However, our notion of plurisubharmonicity works well and has the property that ω-plurisubharmonic functions are subharmonic on all pseudo-holomorphic curves.
We note that on a Gromov manifold there exists a class of Lagrangian plurisubharmonic functions with many good properties. For example, they are subharmonic when restricted to any Lagrangian submanifold which is minimal. This is explored in a separate paper [HL 5 ].

The φ-Hessian.
In this section we prove Theorem 1.4 and Proposition 1.7. The arguments will involve ideas and notation important for the rest of the paper. A generalization of Theorem 1.4 to submanifolds which are φ-critical can be found in Appendix A.
Recall (cf. [ON], p. 86) that the Hessian, or second covariant derivative, of a smooth function f on a riemannian manifold X is defined on tangent vector fields V, W by the Hessian is the symmetric 2tensor, or the symmetric linear map of T x X given by In terms of local coordinate vector fields where Γ k ij are the standard Christoffel symbols of the riemannian connection. Let V be a real inner product space. Given an element φ ∈ Λ p V * , we define a linear map, central to this paper, That is, on simple vectors, one has Recall that the natural inner product on End(V ) is given by: Using this inner product we have the adjoint map which will also be important.
AEÇÌ . If we identify End (V ) with the Lie algebra gl(V ) of GL (V ), then λ φ is the differential of the standard representation of GL(V ) on Λ p V * at φ. Therefore, ker(λ φ ) is the ÁAEÁÌÁÇAE 2.1. The φ-Hessian of a function f ∈ C ∞ (X) is the p-form H φ (f ) defined by letting the symmetric endomorphism Hessf act on φ as a derivation, i.e., (2.5) In terms of the bundle map λ φ : is the image of the Hessian of f . The second order differential operators dd φ and H φ differ by a pure first order operator. This is the first of the three equations needed to prove Theorem 1.4.
as operators on vector fields (L = the Lie derivative). The right hand side of this formula has a standard extension to all tensor fields as a derivation that commutes with contractions. It is zero on functions, that is, it is a bundle endomorphism whose value on T * X is minus the transpose of its value on T X. In particular, we find that Many of the nice results for the dd φ -operator continue to hold in the non-parallel case after replacing it with the φ-Hessian. Perhaps even more importantly, many properties of the dd φ -operator in the parallel case can best be understood by considering the φ-Hessian.
The second formula needed for the proof of Theorem 1.4 is algebraic in nature, involving the bundle map λ φ : End(T X) → Λ p T * X. Consequently, as before, we replace T x X by a general inner product space V . If ξ is a p-plane in V (not necessarily oriented), let P ξ : V → ξ denote orthogonal projection. The following, along with its reinterpretations (2.9) ′ and (2.12), is a central result of this paper. (2.10) Note that if e 1 , ..., e p is an orthonormal basis for the p-plane ξ, then Consequently, it is natural to refer to A, P ξ as the ξ-trace of A and to use the notation In particular, for each A ∈ End(V ), is called a first cousin of ξ. The first cousins of ξ span the tangent space to the Grassmannian G(p, V ) ⊂ Λ p V at the point ξ. Since φ restricted to G(p, V ) is a maximum on G(φ), this fact implies the following result, which we shall use frequently.
Proof of Theorem 2.3. Pick an orthonormal basis for ξ and extend to an orthonormal basis of V . It suffices to prove (2.9) when A = b⊗a with a and b elements of this basis. It is easy to see that b ⊗ a, P ξ = 0 unless a = b ∈ ξ, in which case a ⊗ a, P ξ = 1. By equation is a first cousin of ξ and φ((b ∧ (a l ξ)) = 0 by the First Cousin Principle. If a = b ∈ ξ, then b ∧ (a l ξ) = ξ and therefore φ((b ∧ (a l ξ)) = φ(ξ) = 1.
Theorem 2.3 has many consequences. We mention several. From (2.9) ′ we have: This equation (2.12) is the second equation needed in the proof of Theorem 1.4. Ê Å ÊÃ . Equation (2.12) provides an alternative definition of φ-plurisubharmonic (as well as strictly φ-plurisubharmonic and φ-pluriharmonic) functions, which bypasses the bundle map λ φ and uses only the trace of the Hessian of f on φ-planes ξ.
Another application of Theorem 2.3 is given by: See Remarks A.5 and A.7 for an extension of this to a recent result in [R]. Theorem 2.3 has another useful consequence used to prove Lemma 1.5. Note that for one has e ⊗ e, P ξ ≥ 0, and more generally A, P ξ ≥ 0 whenever A ≥ 0. Since df and ∇f are metrically equivalent, (2.13) Therefore, Theorem 2.3 has the following consequence. (2.14) Theorem 2.3 can also be used to understand the relationship between convex functions and φ-plurisubharmonic functions. A function f ∈ C ∞ (X) is called convex if Hessf ≥ 0 at each point, and it is called affine if Hessf ≡ 0 on X. (If f is affine, ∇f splits X locally as a riemannian product R × X 0 .) ÇÊÇÄÄ Ê 2.8. Every convex function is φ-plurisubharmonic, and every strictly convex function is strictly φ-plurisubharmonic (and every affine function is φ-pluriharmonic).
Ê Å ÊÃ 2.9. The converse always fails; there are always φ-plurisubharmonic functions which are not convex. To see this, consider first the euclidean case with X = V and φ parallel. Recall that the orthogonal projections P e onto lines in V generate the extreme rays of the convex cone of convex functions (positive semi-definite quadratic forms) in Sym 2 V ⊂ End (V). This cone is self-dual. The projections P ξ = λ * φ (ξ) for ξ ∈ G(φ) generate a proper convex subcone (in fact a proper convex subcone of the cone generated by orthogonal projections onto p-planes). Hence, by the Bipolar Theorem there must exist a non-convex quadratic function Q ∈ Sym 2 V with Q, P ξ ≥ 0 for all ξ ∈ G(φ). By (2.9), Q is φ-plurisubharmonic. (Recall that for a convex cone C ⊂ R n with vertex at the origin, the Bipolar Theorem states that (C 0 We now recall some elementary facts about submanifolds. Given a submanifold X ⊂ X, let (•) T and (•) N denote orthogonal projection of T x X onto the tangent and normal spaces of X respectively. Then the canonical riemannian connection ∇ of the induced metric on X is given by This is a symmetric bilinear form on T X with values in the normal space. Its trace H = trace B is the mean curvature vector field of X, and X is called a minimal submanifold if H ≡ 0. Finally, let ∆ denote the Laplace-Beltrami operator on X and Hess denote the Hessian operator on X. The proof of the following is straightforward.
With a change of notation, this is the final formula needed to prove Theorem 1.4.
For future reference we add a remark. Ê Å ÊÃ 2.12. When φ is harmonic, the operator d φ can be expressed in terms of the Hodge d * -operator as and therefore so that the first equation holds if φ is a harmonic form, and in particular if φ is parallel. Note also that for ψ = * φ and Ê Å ÊÃ 2.13. (Examples). Corollary 2.5 gives us the following basic fact.
This infinitesimal version of Theorem 1.4 gives insight into the condition of φ-plurisubharmonicity. Consider for example the calibration where (X, ω) is a Kähler, or more generally a Gromov, manifold. In this case is exactly the set of complex p-planes in T X. Thus a function f is Ω p -plurisubharmonic if and only if tr ξ Hess f ≥ 0 for all complex p-planes ξ. In the Kähler case this means that f is Ω p -plurisubharmonic if and only if it is subharmonic on all p-dimensional complex submanifolds. This condition can be expressed somewhat differently as follows. Any symmetric endomorphism A : T X → T X can be decomposed as is hermitian symmetric with real eigenvalues λ 1 ≤ λ 2 ≤ · · · ≤ λ n . If ξ is a complex p-plane, then tr ξ A = tr ξ A sym . Moreover, the infimum of tr ξ A sym over such planes is 2(λ 1 + · · · + λ p ). From this it follows that a function f is Ω p -plurisubharmonic if and only if the eigenvalues λ 1 ≤ · · · ≤ λ n of its hermitian symmetric hessian {Hess f } sym satisfy λ 1 + · · · + λ p ≥ 0 at all points of X.
There is a parallel story on a quaternionic Kähler manifold (X, Ψ) where Ψ is locally of the form Ψ = 1 6 {ω 2 I + ω 2 J + ω 2 K } for orthogonal almost complex structures I, J, K satisfying the standard relations. For the calibration is the set of quaternionic p-planes in T X. Each symmetric endomorphism A has a quaternionic hermitian symmetric part A qsym = 1 4 (A − IAI − JAJ − KAK) with real eigenvalues λ 1 ≤ · · · ≤ λ n , and a function f is Ψ p -plurisubharmonic if and only if the eigenvalues of {Hess f } qsym satisfy λ 1 + · · · + λ p ≥ 0 at each point.
In Section 6 we show that on a general (X, φ) the squared distance to any φ-free submanifold M is strictly φ-plurisubharmonic on a neighborhood of M . This constructs huge families of φ-plurisubharmonic functions with topologically interesting level sets.

Elliptic Calibrations
In this paper we primarily restrict attention to C ∞ -functions. However, in this section we give the foundations for a theory of more general φ-plurisubharmonic functions developed in [HL 4 ]. Suppose (X, φ) is a calibrated manifold and assume that throughout this section that G(φ) is a fibre bundle over X. ÁAEÁÌÁÇAE 3.1. Given an everywhere positive definite section A of Sym 2 (TX), the associated differential operator will be called a Laplacian on X.
The standard riemannian Laplacian is associated with the identity section of Sym 2 (TX). For each Laplacian ∆ on X, classical potential theory is applicable.
The proofs of these two results will be given in the case φ ∈ Λ p V is a parallel calibration on euclidean n-space. Arguments for the more general case are essentially the same.
Let P + denote the convex cone in Let P + denote the polar cone. That is, H ∈ P + if and only if The Bipolar Theorem (cf. [S]) states that A ∈ Sym 2 (V) can be expressed as in (3.2) if and only if H, A ≥ 0 for all H ∈ P + (3.4) Ä ÅÅ 3.6. Given A ∈ Sym 2 (V), the associated operator ∆ A is a mollifying Laplacian for φ if and only if 1) A = N j=1 λ j P ξ j with λ j > 0 and ξ j ∈ G(φ), and 2) ξ 1 , ..., ξ N involve all the variables, i.e., v l ξ j = 0, j = 1, ..., N implies v = 0.
Proof. Suppose 1) and 2) are valid. Then Av, v = λ j |v l ξ j | 2 and 2) implies that A is positive definite. Moreover, Conversely, suppose ∆ A is a mollifying Laplacian for φ. Take f to be a quadratic function with H = Hessf ∈ P + , so that f is φ-plurisubharmonic. Then ∆ A f = H, A ≥ 0 for all such H. As noted above (3.4) implies (3.2). Finally, note that Av, v = N j=1 λ j |v l ξ j | 2 and therefore, since A is positive definite 2) is verified.
Proof of Theorem 3.4 Suppose there exists a mollifying Laplacian ∆ A for φ. Then by Lemma 3.6 1), we have A = N j=1 λ j P ξ j , and by 2) we have that given v = 0, there exists ξ j ∈ G(φ) with v l ξ j = 0. Thus φ is elliptic.
Conversely, if φ is elliptic, then by compactness, there exists a finite number of ξ j ∈ G(φ) such that A = N j=1 P ξ j ∈ P + is positive definite, thereby verifying 1) and 2).

Generalized φ-plurisubharmonic Functions
Throughout this subsection we assume that φ is an elliptic calibration. The differential operator Hessf extends to distributions f on X via duality producing a well-defined distributional section Hessf of the bundle Sym 2 (TX). By definition, a distributional section of a vector bundle E → X is a continuous linear functional on the space of smooth compactly supported sections of E * ⊗ Λ n T * X, or equivalently, on the space of s ≡ s ⊗ * 1 for s ∈ Γ cpt E * .
for every smooth section ξ of G(φ) and every smooth compactly supported non-negative multiple λ of the volume form on X.
ÅÈÄ . The fundamental function E(x) in Proposition 1.7 defines an L 1 loc (R n ) function and, hence, is a distribution on R n . It is φ-plurisubharmonic for any calibration φ on R n of degree p. This is easy to prove since (for p < n) the distributional hessian Theorem 3.5 extends from functions f ∈ C ∞ (X) to distributions f ∈ D ′ (X).
Proof. Assume that A is of the form A = N j=1 λ j P ξ j with λ j > 0 smooth and each ξ j a smooth section of G(φ). Then λ j tr ξ j Hessf is a well defined distribution on X, and, by hypothesis, it pairs with every smooth, compactly supported non-negative multiple λ of the volume form, to give (∆ A f )(λ) ≥ 0. Hence ∆ A f is a non-negative regular Borel measure on X. The proof of the converse is similar to that of Theorem 3.5. This theorem has a multitude of corollaries, deducible from the classical potential theory for ∆ A . We list just two of the facts.
1) Each ∆ A -subharmonic distribution (and therefore each φ-plurisubharmonic distribution) belongs to L 1 loc (X), the space of locally Lebesgue integrable functions on X. 2) Each ∆ A -subharmonic distribution (and therefore each φ-plurisubharmonic distribution) has a canonical classical representative defined by f dvol which is [−∞, ∞)-valued and upper semi-continuous on X. Here B r (x) denotes the ball of radius r about x and |B r (x)| denotes its volume.
See [HL 4 ] for a development of upper semi-continuous φ-plurisubharmonic functions using these results.

Convexity in Calibrated Geometries
We suppose throughout this section that (X, φ) is a non-compact, connected calibrated manifold and all φ-plurisubharmonic functions are of class C ∞ .
Suppose K is a compact subset of X. Then x / ∈ K if and only if there exists a smooth non-negative φ-plurisubharmonic function f on X which is identically zero on a neighborhood of K and has f (x) > 0. Furthermore, if there exists a φ-plurisubharmonic function on X which is strict at x, then f can be chosen to be strict at x.
Note: One sees easily that K = K. Therefore, if K is compact, the function f in Lemma 4.2 can be taken to be zero on a neighborhood of K (since one can replace K with K).
2) There exists a φ-plurisubharmonic proper exhaustion function f on X.
ÁAEÁÌÁÇAE 4.4. If the equivalent conditions of Theorem 4.3 are satisfied, then (X, φ) is a convex calibrated manifold and X is φ-convex . Proof that 2) ⇒ 1). If K is compact, then c ≡ sup K f is finite and K is contained in the compact pre-level set {x ∈ X : f (x) ≤ c}.
Proof that 1) ⇒ 2). A φ-plurisubharmonic proper exhaustion function on X is constructed as follows. Choose an exhaustion of X by compact (X, φ)-convex subsets K 1 ⊂ K 2 ⊂ K 3 ⊂ · · · with K m ⊂ K 0 m+1 for all m. By Lemma 4.2 and the compactness of K m+2 − K 0 m+1 , there exists a φ-plurisubharmonic function f m ≥ 0 on X with f m identically zero on a neighborhood of K m and f m > 0 on K m+2 − K 0 m+1 . By re-scaling we may assume f m > m on K m+2 − K 0 m+1 . The locally finite sum f = ∞ m=1 f m satisfies 2). Ä ÅÅ 4.5. Condition 2) in Theorem 4.3 is equivalent to the a priori weaker condition: 2) ′ There exists a continuous proper exhaustion function f on X which is smooth and φ-plurisubharmonic outside a compact subset of X.
In fact if f satisfies 2) ′ , then f can be modified on a compact subset to be φ-plurisubharmonic on all of X. Consequently, if f satisfies 2) ′ and is strict outside a compact set, then its modification is also strict outside a compact set.
Proof. For large enough c, f is smooth and φ-plurisubharmonic outside the compact set {x ∈ X : f (x) ≤ c − 1}. Pick a convex increasing function ϕ ∈ C ∞ (R) with ϕ ≡ c on a neighborhood of (−∞, c − 1] and ϕ(t) = t on (c + 1, ∞). Then by Lemma 1.5 the composition ϕ • f is φ-plurisubharmonic on all of X (in particular smooth) and equal to f outside of the compact set {x ∈ X : f (x) ≤ c + 1}.
2) There exists a strictly φ-plurisubharmonic proper exhaustion function for X.
ÁAEÁÌÁÇAE 4.7. If the equivalent conditions of Theorem 4.6 are satisfied, then (X, φ) is a strictly convex calibrated manifold or X is strictly φ-convex.
Proof of Theorem 4.6. Suppose that X is equipped with both a φ-plurisubharmonic proper exhaustion function f and a strictly φ-plurisubharmonic function g. Then the sum f + e g is a strictly φ-plurisubharmonic exhaustion function. Now Theorem 4.6 follows immediately from Theorem 4.3.
We shall construct many φ-convex manifolds in the course of our discussion (See, in particular, §6). However, we present some elementary examples here.
Then dd φ f = pφ and hence f is a strictly φ-plurisubharmonic exhaustion. That is, (R n , φ) is a strictly convex calibrated manifold. ÅÈÄ 2. Suppose φ = dx 1 ∧ · · · ∧ dx n on a domain X in R n . Then dd φ f = (∆f )φ and f is φ-plurisubharmonic if and only if f is subharmonic. Recall that if K ⊂⊂ X, then K = K ∪ {all the "holes" in K relative to X}, (connected components of X − K which are relatively compact in X). Thus (X, φ) is strictly convex for any open set X ⊂ R n .
It is instructive to extend this elementary example. ÅÈÄ 3. Suppose φ = dx 1 ∧ · · · ∧ dx p on a domain X in R n with coordinates (x 1 , ..., x p , y 1 , ..., y n−p ). A function f ∈ C ∞ (X) is φ-plurisubharmonic if and only if ∆ x f ≥ 0 on X. For a set K ⊂ R n , let K y denote the horizontal slice {x ∈ R p : (x, y) ∈ K} of K. Suppose that for each y ∈ R n−p , the horizontal slice X y has no holes in R p . Then (X, φ) is strictly convex. To prove this fact, it suffices to exhaust X by compact sets K with the same property and show that each such K is equal to its (X, φ)-hull. Suppose z 0 = (x 0 , y 0 ) ∈ X − K. Since x 0 is not in a hole of K y 0 in R p , we may choose (by Example 2) an entire subharmonic function g(x) with g(x 0 ) >> 0 and sup K y 0 g << 0. Now pick ψ ∈ C ∞ cpt ({y : |y − y 0 | < ǫ}) with 0 ≤ ψ ≤ 1 and ψ(y 0 ) = 1. Then f (x, y) = g(x)ψ(y) is φ-plurisubharmonic and f (z 0 ) = g(x 0 ) >> 0. For ǫ sufficiently small, sup K f ≤ 0. This proves z 0 does not belong to the (X, φ)-hull of K. ) of X is easily seen to contain the origin. Similarly, a domain of "U"-shape, whose upper boundary along the bottom has a flat segment, is not φ-convex even though it is locally φ-convex (by Example 3).
It is important to "weaken" this notion of strict convexity. ÌÀ ÇÊ Å 4.8. The following two conditions are equivalent: 1) K ⊂⊂ X ⇒ K ⊂⊂ X, and there exists a strictly φ-plurisubharmonic function defined outside a compact subset of X 2) There exists a φ-plurisubharmonic proper exhaustion function on X which is strict outside a compact subset of X. ÁAEÁÌÁÇAE 4.9. If the equivalent conditions of Theorem 4.8 are satisfied, then the calibrated manifold (X, φ) is strictly convex at ∞ or X is strictly φ-convex at ∞. Ê Å ÊÃ . This is not the standard terminology used in complex geometry where such spaces are called "strongly (pseudo) convex".
2) ′ There exists a continuous proper exhaustion function f on X which is smooth and strictly φ-plurisubharmonic outside a compact subset of X.
By Lemma 4.5, Condition 2) ′ implies Condition 2). Now assume 1). Since K ⊂⊂ X implies K ⊂⊂ X, we know from Theorem 4.3 that there exists a φ-plurisubharmonic exhaustion function f for X. Let g denote the strictly φplurisubharmonic function which is only defined outside of a compact set. We can assume this compact set is {x ∈ X : f (x) ≤ c} for some large c. Then h ≡ max{f + e g , c} is a continuous proper exhaustion function which, outside the compact set {x ∈ X : f (x) ≤ c}, is strictly φ-plurisubharmonic (in fact, equal tof + e g ). This proves 2) ′ and completes the proof of the theorem.

Cores.
In each non-compact calibrated manifold (X, φ) there are certain distinguished subsets which play an important role in the φ-geometry of the space. (In complex manifolds which are strongly pseudoconvex, these sets correspond to the compact exceptional subvarieties.) The remainder of this section is devoted to a discussion of these subsets.
ÁAEÁÌÁÇAE 4.11. The core of X is defined to be the intersection Core(X) ≡ W (f ) over all f ∈ PSH(X, φ). The inner core of X is defined to be the set InnerCore(X) of points x for which there exists y = x with the property that f (x) = f (y) for all f ∈ PSH(X, φ).
Proof. If x / ∈ Core(X), then there exists g ∈ PSH(X, φ) with g strict at x. Suppose y = x. Then if ψ is compactly supported in a small neighborhood of x missing y, and ψ has sufficiently small second derivatives, one has f = g + ψ ∈ PSH(X, φ). Obviously for such f , the values f (x) and f (y) can be made to differ, so therefore x / ∈ InnerCore(X).
ÈÊÇÈÇËÁÌÁÇAE 4.13. Every compact φ-submanifold M without boundary in X is contained in the inner core.
Proof. Each f ∈ PSH(X, φ) is subharmonic on M by Theorem 1.4. Hence, f is constant on M .
ÈÊÇÈÇËÁÌÁÇAE 4.14. Suppose X is φ-convex. Then Core(X) is compact if and only if X is strictly φ-convex at ∞, and Core(X) = ∅ if and only if X is strictly φ-convex .
Proof. If X is strictly φ-convex at ∞, then choosing f to satisfy 2) in Theorem 4.8, we see that the Core(X) ⊂ W (f ) is compact. Obviously, strict φ-convexity implies that Core(X) = ∅. Conversely, if Core(X) is compact, then in the construction of the φ-plurisubharmonic exhaustion function in the proof of Theorem 4.3 we may choose K 1 to be the φ-convex hull of Core(X). Then by the definition of Core(X) and Lemma 4.2, each of the functions f m in that proof can be chosen to be strictly φ-plurisubharmonic on K m+2 − K 0 m+1 . Hence the exhaustion f = m f m is strictly φ-plurisubharmonic outside a compact set containing the core.
A slight modification of this construction gives the following general result. ÈÊÇÈÇËÁÌÁÇAE 4.15. Suppose X is strictly φ-convex at ∞, and K ⊂ X is a compact, φconvex subset containing the core of X. Let U be any neighborhood of K. Then there exists a proper φ-plurisubharmonic exhaustion function f : X → R + which is strictly φ-plurisubharmonic on X − U , and identically zero on a neighborhood of K.
Proof. Choose K 1 as in the construction of the φ-plurisubharmonic exhaustion function given in the proof of Theorem 4.3. Let K(ǫ) denote the compact ǫ-neighborhood of K 1 . Then , then for each f ∈ PSH(X, φ), we have f (x) ≤ sup K(ǫ) f . However, inf ǫ sup K(ǫ) f = sup K f , and we conclude that x ∈ K. Thus we can choose K 2 ≡ K(ǫ) in our construction of f , and for small enough ǫ we have K 2 ⊂ U as well as K 1 ⊂ K 0 2 . The proof is now completed as in the proof of Proposition 4.14.

Examples of Complete Convex Manifolds and Cores
In §6 (Theorem 6.4) we shall show that there are many strictly φ-convex domains in any calibrated manifold (X, φ). They can have quite arbitrary topological type within the strictures imposed by Morse Theory and φ-positivity of the Hessian. However, it is also interesting geometrically to ask for convex manifolds which are complete.
In fact, there exist enormous families of complete calibrated manifolds (X, φ) with ∇φ = 0 which are strictly φ-convex at infinity. For example, any (X, φ) which is asymptotically locally euclidean (ALE) is such a creature. In this case the radial function on the asymptotic cone at infinity is strictly convex. It appears likely the corresponding assertion also holds for quasi ALE spaces. For the general construction of such spaces with SU(n), Sp(n), G 2 , or Spin 7 holonomy, the reader is referred to the book of Joyce [J].
Certain manifolds of this type have been quite explicitly constructed, and in these cases one can explicitly construct φ-plurisubharmonic exhaustion functions and identify the cores. We indicate how to do this below.
We begin however with an observation in dimension 4. Every crepant resolution of singularities of C 2 /Γ, for a finite subgroup Γ ⊂ SU(2), admits a Ricci-flat ALE Kähler metric. On each such manifold there exists an S 2 -family of parallel calibrations where ω is the given Kähler form, ϕ = Re{Φ} and ψ = Im{Φ} and Φ is a parallel section of the canonical bundle κ X . Let E = π −1 (0) be the exceptional locus of the resolution. Then for any φ ∈ C we have This follows from the fact that each φ ∈ C is in fact the Kähler form for a complex structure on X compatible with the given metric. With this complex structure X is pseudo-convex, and by the Stein Reduction Theorem (cf. [GR,p. 221]) we know its core is the union of its compact complex subvarieties. For φ = ω there are no such subvarieties since by the Wirtinger inequality (cf. [L 1,2 ]), applied to φ, they would necessarily be homologically mass-minimizing, and by the same result applied to ω any such subvariety is ω-complex (and therefore a component of E). ÅÈÄ 1. (Calabi Spaces). Let X → C n /Z n be a crepant resolution of C n /Z n where the action on C n is generated by scalar multiplication by τ = e 2πi/n . Following Calabi [C] we define the function F : C n /Z n → R by F (ρ) = n ρ n + 1 + 1 n n−1 k=0 τ k log n ρ n + 1 − τ k where ρ ≡ z 2 (pushed down to C n /Z n ), and the log is defined by choosing arg ζ ∈ (−π, π). We then define a Kahler metric on C n /Z n − {0} by setting Calabi shows that this metric is Ricci-flat and (when pulled back) extends to a Ricci flat metric on X. The parallel form Φ = dz 1 ∧ · · · ∧ dz n extends to a parallel section of κ X . This metric is given explicitly on R 2n /Z n by where φ = Re{Φ}. Hence, X is a complete, strictly φ-convex manifold. is strictly ϕ − plurisubharmonic on S − Z where ϕ denotes the associative calibration on S. Since Z is an associative submanifold we conclude that Core(S) = Z.
In an analogous fashion the authors construct a complete riemannian metric with Spin 7holonomy on the total space S of a spinor bundle over S 4 . (See [BS,page 847,Case ii].) A similar calculation shows that there exists an exhaustion function which is strictly Φplurisubharmonic on S − Z where Φ denotes the Cayley calibration Z the zero-section of S. Since Z is a Cayley submanifold, we conclude that Core( S) = Z.

5.3.
Assume φ ∈ Λ p R n is a calibration. Suppose ∂Ω is (strictly) φ-convex in R n , and locally near a point p ∈ ∂Ω, let ∂Ω be graphed over its tangent space by a function x n = u(x ′ ) for linear coordinates (x ′ , x n ) on R n . Then each nearby hypersurface: The next lemma and its corollary will be used to establish the main results of this section. Ä ÅÅ 5.4. Suppose ρ is a smooth real-valued function on a riemannian manifold, and ψ : R → R is smooth on the image of ρ. Then for all oriented tangent p-planes ξ.
Although a general defining function for a strictly φ-convex boundary may not be φ-plurisubharmonic, for some applications the following is useful. ÈÊÇÈÇËÁÌÁÇAE 5.7. Suppose Ω ⊂⊂ X has strictly φ-convex boundary ∂Ω with defining function ρ. Then, for A sufficiently large, the function ρ ≡ ρ + Aρ 2 is strictly φ-convex in a neighborhood of ∂Ω and also a defining function for ∂Ω.
One might hope for a converse to Theorem 5.6, e.g., if the domain Ω is φ-convex then the boundary is φ-convex. However, the following elementary example shows that this is false.

ÅÈÄ
. Let φ ≡ dx ∧ dy in R 3 as in Example 3 of section 4. Let X denote the solid torus obtained by rotating the disk {(y, z) : y 2 + (z − R) 2 < r 2 } about the y-axis. Since each slice X z has no holes in R 3 , the domain X is φ-convex (cf. Example 3 of §4). However, the boundary torus ∂X is φ-convex if and only if 2r ≤ R. This follows from an elementary calculation which uses the obvious defining function and Definition 5.1 (or by using Proposition 5.13 below) ÉÍ ËÌÁÇAE 5.8. For which strictly convex calibrated manifolds is it true that φ-convex subdomains have φ-convex boundaries? More generally, when is the φ-convexity of a domain a local condition at the boundary?

The Second Fundamental Form
The φ-convexity of a boundary can be equivalently defined in terms of its second fundamental form. Note that if M ⊂ X is a smooth hypersurface with a chosen unit normal field n we have a quadratic form II defined on T M by II(V, W ) = B V,W , n where B denotes the second fundamental form of M discussed in §2. For example, when H = S n−1 (r) ⊂ R n is the euclidean sphere of radius r, oriented by the outward-pointing unit normal, we find that II(V, W ) = − 1 r V, W . For the sake of completeness we include a proof of the following standard fact. Ä ÅÅ 5.11. Suppose ρ is a defining function for Ω and let II denote the second fundamental form of the hypersurface ∂Ω oriented by the outward-pointing normal. Then Hess ρ T ∂Ω = −|∇ρ| II and therefore tr ξ Hessρ = −|∇ρ| tr ξ II for all ξ ∈ G(p, T ∂Ω).
Ê Å ÊÃ . Recall that a defining function ρ for Ω satisfies |∇ρ| ≡ 1 in a neighborhood of ∂Ω if and only if ρ is the signed distance to ∂Ω (< 0 in Ω and > 0 outside of Ω). In fact any function ρ with |∇ρ| ≡ 1 in a riemannian manifold is, up to an additive constant, the distance function to (any) one of its level sets. In this case it is easy to see that where II denotes the second fundamental form of the hypersurface H = {ρ = ρ(x)} with respect to the normal n = ∇ρ and the blocking in (5.7) is with respect to the splitting T x X = span (n x ) ⊕ T x H. For example let ρ(x) = x ≡ r in R n . Then direct calculation shows that Hess ρ = 1 r (I −x •x) wherex = x/r.
As an immediate consequence we have ÈÊÇÈÇËÁÌÁÇAE 5.13. Let Ω ⊂ X be a domain with smooth boundary ∂Ω oriented by the outward-pointing normal. Then ∂Ω is φ-convex if and only if its second fundamental form satisfies tr ξ II ≤ 0 for all φ-planes ξ which are tangent to ∂Ω. This can be expressed more geometrically by saying that tr B ξ must be inward − pointing for all tangential φ-planes ξ.
We finish this section with a useful characterization of strictly convex domains.
ÌÀ ÇÊ Å 5.14. Let (X, φ) be a strictly φ-convex manifold and Ω ⊂⊂ X a domain with smooth boundary. Then the following are equivalent.
There exists a defining function ρ ∈ C ∞ (Ω) for ∂Ω which is strictly φ-plurisubharmonic on a neighborhood of Ω.

φ-Free Submanifolds and Topology
Somewhat surprisingly, for any calibration φ there is a precise integer bound on the homotopy dimension of any strictly φ-convex domain. This is the first result below. After establishing it, we show that on the other hand, subject to this bound, there exist strictly convex domains of almost arbitrary topological type. ÁAEÁÌÁÇAE 6.1. The free dimension, denoted fd(φ), of a calibrated manifold (X, φ) is the maximum dimension of a linear subspace in T X which contains no φ-planes. Such subspaces will be called φ-free.
Proof. Let f : X → R + be a strictly φ-plurisubharmonic proper exhaustion function. By perturbing we may assume that f has non-degenerate critical points. The theorem follows if we show that each critical point has Morse index ≤ fd(φ) (cf. [M]). If this fails, then there is a critical point x at which Hess x f has at least fd(φ) + 1 negative eigenvalues. In particular, there exists a subspace W ⊂ T x X of dimension = fd(φ) + 1 with Hess x f W < 0. However, by definition of fd(φ), W must contain a φ-plane ξ ∈ G(φ), and since f is strictly φ-convex , we must have tr ξ Hess x f > 0, a contradiction.

ÇÅÅ AEÌË
For (a), note that every real subspace of dimension n + 1 in C n contains a complex line and is therefore not free. The free subspaces of dimension n are exactly the totally real n-planes -those for which JW ∩ W = {0}.
For the second statement, recall that the φ-planes are exactly the J-invariant subspaces of dimension 2p. Now if W ⊂ C n has codimension ≤ n − p, then dim R {W ∩ JW } ≥ 2p, and so W is not φ-free. However for a generic W of codimension n − p + 1, the maximal complex subspace of W satisfies dim R {W ∩ JW } = 2p − 2.
For (b), we first show that every real hyperplane H ⊂ C n contains a Special Lagrangian n-plane and is therefore not free. Choose a unit vector n ⊥ H and consider the orthogonal decomposition C n = (Rn) ⊕ We now observe that for a real subspace W ⊂ C n of dimension 2n − 2, For the first equivalence note that if J(W ⊥ ) ⊂ W , then the construction above gives a Special Lagrangian L ⊂ W . Conversely, given L ⊂ W , For the second statement we use that fact that G(Ψ p ) is exactly the set of quaternionic linear subspaces of quaternionic dimension p in H n . The argument then proceeds as in part two of (a).
For (d), suppose V ⊂ ImO has dimension 5. Let x, y be an orthonormal basis of V ⊥ . Choose any unit vector ǫ ∈ V perpendicular to z ≡ xy. Then span {z, ǫ, zǫ} ⊂ V is associative, and so V is not free. (To see that zǫ ∈ V , note that left multiplication by z is an isometry which preserves the quaternion subalgebra span{x, y, z} and therefore also preserves its orthogonal complement.) We now claim that To see this, suppose there exists an associative 3-plane V ⊂ W . Then V ⊥ = Rǫ ⊕ W ⊥ is coassociative (where ǫ ∈ W is a unit vector perpendicular to V ). Choose an orthonormal basis x ′ , y ′ , z ′ of W ⊥ . Coassociativity is equivalent to the fact that the 4form ( * φ)(ǫ, x ′ , y ′ , z ′ ) = ǫ, [x ′ , y ′ , z ′ ] = ±1 where [·, ·, ·] is the associator. We now recall the general equality φ(x ′ , y ′ , z ′ ) 2 + [x ′ , y ′ , z ′ ] 2 = 1 (cf. [HL 1 ]) from which it follows that φ(x ′ , y ′ , z ′ ) = 0, that is, W ⊥ is φ-isotropic. Conversely, supposing W ⊥ is φ-isotropic, the equality shows that [x ′ , y ′ , z ′ ] is a unit vector and therefore U ≡ span {x ′ , y ′ , z ′ , [x ′ , y ′ , z ′ ]} is coassociative. Hence, U ⊥ ⊂ W is associative. For (e), suppose V ⊂ ImO has dimension 5. Let x, y be an orthonormal basis of V ⊥ . Then U = span {x, y, xy} is associative, and so U ⊥ ⊂ V is coassociative. Hence, V is not free. Of course the free 4-planes are exactly those which are not coassociative.
For (f), suppose V ⊂ O has dimension 5. Let x, y, z be an orthonormal basis of V ⊥ . Then W = span {x, y, z, x × y × z} is a Cayley plane, and so W ⊥ ⊂ V is also Cayley. Hence, V is not free. The free 4-planes are exactly those which are not Cayley.
We now show that within the homotopy restrictions imposed by Theorem 6.2, the possible topologies for strictly φ-convex manifolds are vast.
Let (X, φ) be a calibrated manifold. A p-plane ξ is said to be tangential to a submanifold M ⊂ X if span ξ ⊂ T x M . ÁAEÁÌÁÇAE 6.3. A closed submanifold M ⊂ X is φ-free if there are no φ-planes ξ ∈ G(φ) which are tangential to M . If the restriction of the calibration φ to M vanishes, M is called φ-isotropic.
Note that φ-isotropic submanifolds are φ-free. Each submanifold of dimension strictly less than the degree of φ is φ-isotropic and hence automatically φ-free. Furthermore, in dimension p the generic local submanifold is φ-free. Depending on the geometry, this may continue through a range of dimensions greater than p. Proof. We begin with the following. Ä ÅÅ 6.5. Fix x ∈ M and let P N : T x X → N denote orthogonal projection onto the normal plane of M at x. Then for each ξ ∈ G(φ) one has for any function f . The lemma then follows from the assertion that To see this we first note that the Hessian of any function f can be written for all V, W ∈ T x X. It follows that if ∇f = 0 on the submanifold M , then T x M ⊂ Null(Hess x f ). Thus, with respect to the decomposition T x X = T x M ⊕ N we have and it remains to show that A is the identity. To see this, set δ(x) = dist M (x) and note that ∇δ = n is a smooth unit-length vector field near (but not on) M whose integral curves are geodesics emanating from M . Hence, ∇ n (∇f M ) = ∇ n (∇ 1 2 δ 2 ) = ∇ n (δn) = n + δ∇ n n = n.
Taking limits along normal geodesics down to M gives the result.
Theorem 6.4 now follows from the fact that P N , P ξ ≥ 0 with equality iff span ξ ⊂ N ⊥ = T x M.
The following result gives us a powerful, very general method for constructing strictly φ-convex domains in (X, φ). ÌÀ ÇÊ Å 6.6. Suppose M is a φ-free submanifold of (X, φ). Then there exists a fundamen- Ê Å ÊÃ . The existence of φ-free submanifolds gives the existence of strictly φ-convex domains with essentially every topological type permitted by Morse Theory (Theorem 6.2). Note in particular that if M ⊂ X is φ-free, then every submanifold of M is also φ-free.
Proof. We construct tubular neighborhoods of M as follows. Let ǫ ∈ C ∞ (M ) be a smooth function which vanishes at infinity and has the property that for each x ∈ M the ball {y ∈ X : 1 2 dist(y, x) 2 ≤ ǫ(x)} is compact and geodesically convex. Assume also that ǫ is sufficiently small so that the exponential map gives a diffeomorphism exp : N ǫ −→ U ǫ from the open set N ǫ in the normal bundle N defined by 1 2 n x 2 < ǫ(x) to the neighborhood of M in X. Each U ǫ admits a deformation retraction onto M . By Theorem 6.4 the function f M = 1 2 dist 2 M (·) is strictly φ-plurisubharmonic on a neighborhood of M , which we can assume to be W . We impose the following additional condition on the function ǫ ∈ C ∞ (W ). f M − tǫ is strictly φ − plurisubharmonic on W for 0 ≤ t ≤ 1.
(6.6) Since (6.6) is valid as long as ǫ and its first and second derivatives vanish sufficiently fast at infinity, it is easy to see that the family F (M ) of neighborhoods U ǫ constructed above with ǫ satisfying (6.6) is a fundamental neighborhood system for M . Obviously, the function ψ ≡ (ǫ − f M ) −1 is a proper exhaustion for U ǫ . Recall that if g is a positive concave function, then 1/g is convex, or more directly, calculate that (6.7) Applying λ φ to (6.7) proves that (ǫ − f M ) −1 is strictly φ-plurisubharmonic on {f M < ǫ} = U ǫ . Hence, U ǫ is strictly φ-convex.
To prove parts (b) and (c) one uses Proposition 3.16 in [HL 2 ], characterizing denseness of PSH (V, φ) in PSH (U, φ) in terms of relative convexity, and argues exactly as on page 302 of [HW 1 ]. ÅÈÄ 6.7. As mentioned above, Theorem 6.6 exhibits a rich family of φ-convex domains in (X, φ). For example, let M ⊂ X be any submanifold of dimension < p = degφ. Then by 6.6, M has a fundamental system of neighborhoods each of which is a strictly φ-convex domain homotopy equivalent to M . ÅÈÄ 6.8. Interesting examples occur in all the calibrated geometries examined in depth in [HL 1 ]. Suppose for instance that X is a Calabi-Yau manifold with Special Lagrangian calibration φ. Then any complex submanifold Y ⊂ X (of positive codimension) is φ-free. It follows that any smooth submanifold of Y is also φ-free.

Zero Sets of Non-negative Strictly φ-plurisubharmonic Functions
We now consider the following two classes of subsets of (X, φ).
(2) Zero sets of non-negative strictly φ-plurisubharmonic functions f . These two classes are basically the same, as described in the following two propositions. Choose ψ ∈ C ∞ (X) with ψ ≥ 0 and A = {x ∈ X : ψ = 0}. Now choose ǫ ∈ C ∞ (X) with ǫ(x) > 0 for all x ∈ X, and with ǫ and its derivatives sufficiently small so that f ≡ h + ǫψ is strictly φ-plurisubharmonic on M .
Since ∇ ∂f ∂y 1 , ..., ∇ ∂f ∂y r are linearly independent at x, M is a codimension r submanifold locally near x.
Note that ker(Hess x f ) = T x M . It remains to show that ker(Hess x f ) is φ-free (since if M is φ-free at x, then M is φ-free in a neighborhood of x). This is proved in Lemma 6.11 below. Ä ÅÅ 6.11. Suppose f is strictly φ-plurisubharmonic at x ∈ X. Then ker(Hess Proof. If ker(Hess x f ) ⊂ T x X is not φ-free, there exists ξ ∈ G(φ) with (Hess x f ) span ξ = 0. Consequently, (H φ f )(ξ) = λ φ (Hess x f )(ξ) = tr ξ (Hess x f ) = 0, and f is not strict at x. Ê Å ÊÃ 6.12. Theorem 6.6 can be generalized as follows. Suppose M = {f = 0} is the zero set of a non-negative strictly φ-plurisubharmonic function f on (X, φ). Then there exists a fundamental neighborhood system F (M ) of M consisting of strictly φ-convex domains which satisfy (c) of Theorem 6.6. The neighborhoods U ǫ ∈ F (M ) are defined by U ǫ = {x ∈ X : f (x) < ǫ(x)} where ǫ > 0 is a C ∞ function on X vanishing at infinity along with its first and second derivatives so that f − ǫ remains strictly φ-plurisubharmonic . The proofs of (b), (c) and (d) are essentially the same as in Theorem 6.6.
We conclude with the following useful observation. ÈÊÇÈÇËÁÌÁÇAE 6.13. Let M be a submanifold of (X, φ) and f a smooth function defined on a neighborhood of M such that: (1) ∇f ≡ 0 on M , and (2) f is strictly φ-plurisubharmonic at all points of M . Then M is φ-free.
Recall that at a point ξ ∈ G there is a canonical isomorphism: T ξ G ∼ = Hom (span ξ, (span ξ) ⊥ ). (A.1) On the other hand, T ξ G is canonically a subspace of Λ p V . It is exactly the subspace spanned by the first cousins of ξ. More specifically, the isomorphism (A.1) associates to L : span ξ → (span ξ) ⊥ the p-vector D L ξ. ÁAEÁÌÁÇAE A.3. Let A ∈ End(V ) be a linear map. At each point ξ ∈ G we define a tangent vector This vector field ξ → D A ξ on G is called the A-vector field.
Ê Å ÊÃ . A straightforward calculation shows that if A is symmetric, this A-vector field on G is the gradient of the height function F A : G → R given by F A (ξ) = A, P ξ . ÈÊÇÈÇËÁÌÁÇAE A.4. Suppose φ ∈ Λ p V * and A ∈ End (V ). Then for all p-planes ξ ∈ G(p, V ), Proof. Pick an orthonormal basis for ξ and extend to an orthonormal basis of V . It suffices to prove (A.2) when A = b ⊗ a with a and b elements of this basis. Using formula (2.11) we see the following.
We now define an oriented submanifold M of X to be φ-critical if − → T x M ∈ G cr (φ) for all x ∈ M . We leave it to the reader to use Proposition A.2 to establish the following extension of the previous results.
ÌÀ ÇÊ Å A.6. Suppose φ is a p-form on a riemannian manifold X and M ⊂ X is a φ-critical submanifold with mean curvature vector field H. Then for all f ∈ C ∞ (X), . Let φ = 1 6 {ω 2 I + ω 2 J + ω 2 K } be the quaternion calibration on H n . Then ± 1 3 are critical values and the φ-critical submanifolds with critical value ± 1 3 include all complex Lagrangian submanifolds for any complex structure defined by right multiplication by a unit imaginary quaternion (cf. [U]). Ê Å ÊÃ A.7. In a very interesting recent paper Colleen Robles [R] has shown that for any given parallel calibration φ, λ φ (SkewEnd(T X)) generates an exterior differential system whose integral submanifolds are exactly the φ-critical submanifolds. At a point x this can be stated equivalently as follows. Given ξ ∈ G p (T x X) λ φ (A)(ξ) = 0 ∀ A ∈ SkewEnd(T x X) ⇐⇒ ξ ∈ G cr (φ). (A.4) This can be derived from (A.2) and (A.3).
Ê Å ÊÃ B.3. A symmetric n × n-matrix A is ≥ 0 iff A, P ≥ 0 for all rank-one symmetric n × n-matrices P . Proof. Necessity is already done. For sufficiency fix a, b, c. For each ℓ ⊂ span {v, w} let ξ ∈ G(φ) be the oriented p-plane ℓ ⊕ ξ 0 given in the hypothesis, and note that by equation (B.2) where v ℓ = v, e e, w ℓ = w, e e, and ℓ = span {e}. Now the matrix v 2 ℓ v ℓ w ℓ v ℓ w ℓ w 2 ℓ is rank-one, and every rank-one 2 × 2 matrix, up to positive scalars, occurs in this family. The result follows from Remark B.3. ÁAEÁÌÁÇAE B.6. A calibration φ on a manifold X is called rich (or 2-rich) if for any 2-plane P ⊂ T x X at any point x, and for any line ℓ ⊂ P , there exists a (p − 1)-plane ξ 0 ⊂ P ⊥ so that ±ℓ ⊕ ξ 0 is a φ-plane. ÈÊÇÈÇËÁÌÁÇAE B.7. Let (X, φ) be a rich calibrated manifold. Suppose u 1 , u 2 are φ-pluriharmonic functions on X with ∇u 1 ∧ ∇u 2 = 0 on a dense set. Then for any C 2 -function g(t 1 , t 2 ) F = g(u 1 , u 2 ) ∈ PSH(X, φ) ⇔ g is convex Proof. Apply Proposition B.1, equation (B.2) and Lemma B.5. ÈÊÇÈÇËÁÌÁÇAE B.8. The Special Lagrangian calibration on a Calabi-Yau n-fold, n ≥ 3, and the associative and coassociative calibrations on a G 2 -manifold are rich calibrations.
Alternatively, as noted by the referee, these latter cases follow easily from the fact that G 2 acts transitively on the Stiefel manifolds V 2,7 of ordered pairs of orthonormal 2-vectors in R 7 (see for example [HL 1 , Prop. IV.1.10]).
Formula (B.1) ′ can be usefully applied to more general functions u j . For example, in the Special Lagrangian case on C n with φ = Re(dz), one has that dd φ ( 1 2 |z k | 2 ) = φ, for any complex coordinate z k in any unitary coordinate system on C n . Hence a linear combination of these functions has the property that dd φ u = cφ for some constant c. ÈÊÇÈÇËÁÌÁÇAE B.9. Let (X, φ) be a rich calibrated manifold. Suppose u 1 , ..., u n ∈ C ∞ (X) satisfy the equations dd φ u i = c i φ for constants c 1 , ..., c n . Then for any C 2 -function g(t 1 , ..., t n ) F = g(u 1 , ..., u n ) ∈ PSH(X, φ) ⇔ n i=1 c i ∂g ∂t i Id + Hess g , (( (∇u i ) ξ , (∇u j ) ξ )) ≥ 0 for all φ-planes ξ at all points of X.