Geometric plurisubharmonicity and convexity - an introduction
aa r X i v : . [ m a t h . DG ] N ov GEOMETRIC PLURISUBHARMONICITYAND CONVEXITY- AN INTRODUCTIONF. Reese Harvey and H. Blaine Lawson, Jr. ∗ ABSTRACT
This is an essay on potential theory for geometric plurisubharmonicfunctions. It begins with a given closed subset G l of the Grassmann bundle G ( p, T X ) of tangent p -planes to a riemannian manifold X . This determinesa nonlinear partial differential equation which is convex but never uniformlyelliptic ( p < dim X ). A surprising number of results in complex analysiscarry over to this more general setting. The notions of: a G l -submanifold, anupper semi-continuous G l -plurisubharmonic function, a G l -convex domain, a G l -harmonic function, and a G l -free submanifold, are defined. Results includea restriction theorem as well as the existence and uniqueness of solutions tothe Dirichlet Problem for G l -harmonic functions on G l -convex domains. TABLE OF CONTENTS
1. Introduction2. G l -Plurisubharmonicity for Smooth Functions.3. G l -Submanifolds and Restriction.4. G l -Convexity and the Core.5. Boundary Convexity.6. Upper Semi-Continuous G l -Plurisubharmonic Functions.7. G l -Harmonic Functions and the Dirichlet Problem.8. Geometric Subequations Involving all the Variables.9. Distributionally G l -Plurisubharmonic Functions. Appendices:
A. Geometric Subequations.B. The Linear-Geometric Case. ∗ Partially supported by the N.S.F. 1 . Introduction
In a recent series of papers [HL ]–[HL ] the authors have studied certain aspectsof degenerate non-linear elliptic partial differential equations and “subequations”. Theresults include the development of a generalized potential theory, a restriction theorem,and solutions to the Dirichlet Problem. An important special case – and, in fact, themotivating case – of all these results is the “geometric” one, in which the equation isdetermined by a distinguished family G l of tangent p -planes on a manifold (as we explainbelow). There are many interesting geometric cases coming, for instance, from the theoryof calibrations, from almost complex and quaternionic geometry, and from p -convexity inriemannian and hermitian geometry. However, these examples will not be emphasized heresince they occur in profusion in the earlier papers.One aim of this paper is to collect together the various results in the geometric case.Because of their importance as motivation and their usefulness in non-geometric cases, wethought it would be helpful to present them in a coordinated fashion. This exposition alsoincludes several new theorems.Given an n -dimensional riemannian manifold X , let G ( p, T X ) denote the Grassmannbundle whose fibre at a point x is the set of p -dimensional subspaces of the tangent space T x X . The starting point is to distinguish a subset G l ⊂ G ( p, T X ) determining the partic-ular “geometry”. Then, for example, one defines the G l -submanifolds to simply be those p -dimensional submanifolds M of X with T x M ∈ G l for all x ∈ M . There is also the ana-lytical notion of a G l -plurisubharmonic function , defined for smooth functions u by usingthe riemannian hessian Hess x u . For each W ∈ G ( p, T x X ), one can restrict this quadraticform on T x X to W and take its trace. We then define u ∈ PSH ∞ G l ( X ), the set of smooth G l -plurisubharmonic functions on X , by requiring that:tr W Hess x u ≥ ∀ W ∈ G l x , ∀ x ∈ X. (1 . P ( G l x ) ⊂ Sym ( T x X ) of G l -positive quadratic forms (i.e., those satisfying (1.1))is a closed convex cone with vertex at the origin but it is never uniformly elliptic, unless p = dim X .The smooth theory, i.e., the study of PSH ∞ G l ( X ), is for the most part a straightforwardextension of standard results in complex analysis – where G l is simply the set of complexlines in C n , and the functions u ∈ PSH ∞ G l ( X ) are the standard smooth plurisubharmonicson a domain X ⊂ C n . In Section 4 the existence of various kinds of exhaustion functionsfor X are characterized in terms of G l -convex hulls and the G l -core. The G l -core is emptyif and only if X admits a smooth strictly G l -plurisubharmonic function (Definition 4.1and Theorem 4.2). We recall the notion of a G l -free submanifold which generalizes thenotion of a totally real submanifold in complex analysis. The maximal possible dimensionof such submanifolds provides an upper bound on the homotopy type of strictly G l -convexmanifolds (Theorem 4.16). In Section 5 the G l -convexity of the boundary of a domain isdefined and related to the second fundamental form of the boundary, and also to propertiesof local defining functions for the boundary.The notion of G l -plurisubharmonicity for a general upper semi-continuous function u is defined in Section 6 by requiring that each “viscosity” test function ϕ for u at each point2 ∈ X satisfies (1.1) (cf. [C],[CIL]). A key positivity condition (Remark 6.3) is satisfied,which ensures that smooth G l -plurisubharmonic functions are also G l -plurisubharmonicin the second sense (cf. Lemma 6.2). A surprising number of the basic properties ofplurisubharmonic functions in complex analysis carry over to the general geometric case,provided that G l is a closed set which locally surjects onto X (Theorem 6.5).Under the additional (but still quite weak) assumption that G l admits a smoothneighborhood retraction which preserves the fibres of the projection π : G ( p, T X ) → X , restriction holds in the sense that for any upper semi-continuous u ∈ PSH G l ( X ) and anyminimal G l -submanifold M ⊂ X , the restriction u (cid:12)(cid:12) M is subharmonic for the riemannianLaplacian ∆ M on M (Theorem 6.7). That is, u (cid:12)(cid:12) M is subharmonic in any of the many(equivalent) classical senses. For instance, u (cid:12)(cid:12) M is “sub-the-∆ M -harmonics”. Finally, ifeach W ∈ G l is the tangent space to some minimal G l -submanifold M , then the converseto restriction also holds. This justifies the terminology “plurisubharmonic’.Next we discuss the solution to the Dirichlet problem on domains Ω ⊂⊂ X withsmooth strictly G l -convex boundary and no core.A smooth function u is G l − harmonic if in addition to the inequality (1.1) holding,at each point x there exists a W ∈ G l x such that equality holds, i.e., tr W Hess x u = 0. Interms of the set P ( G l x ) defined by (1.1), this is the requirement that Hess x u ∈ ∂ P ( G l x ) ateach point x .The notion of the Dirichlet dual g P ( G l ) of P ( G l ), defined in (7.1), enables one toextend this notion of G l -harmonicity to general continuous functions since ∂ P ( G l ) = P ( G l ) ∩ ( − g P ( G l )) and g P ( G l ) satisfies the positivity condition required of a subequation (seeSection 7). First, we give a proof of the maximum principle for any upper semi-continuousfunction u which is g P ( G l )-subharmonic (much weaker than G l =plurisubharmonic ) underour hypothesis that the G l -core is empty. This easily established result is a precursor tocomparison. This notion of g P ( G l )-subharmonic is referred to as dually G l -plurisubharmonic in this paper.As long as G l is in a weak sense modeled on a euclidean case G l ⊂ G ( p, R n ), bothexistence and uniqueness hold for the Dirichlet Problem for G l -harmonic func-tions on Ω (see Definition 7.5 and Theorem 7.6). An outline of our proof from [HL ] isprovided in Section 7.Since each closed convex set in a vector space V (in our case Sym ( T x X )) is theintersection of its supporting closed half-spaces, linear subequations can be made to playa special role in understanding our G l -subequations. This is seen in Sections 8 and 9.In Section 8 we consider the case where each G l x involves all the variables in thetangent space T x X . This means there does not exist a proper linear subspace W ⊂ T x X with G l x ⊂ Sym ( W ), and it is equivalent (see Lemma 8.1) to the condition that thereexists A ∈ Span G l with A >
0. Under the mild condition of regularity (Definition 6.8),this enables one to write the subequation P ( G l ) locally as the intersection of a family ofuniformly elliptic subequations (Corollary 8.3), a fact that has many consequences. Oneis the Strong Maximum Principle for G l -plurisubharmonic functions (see Theorem 8.5).There is a distributional notion of G l -plurisubharmonicity (but not of G l harmonic-ity). In Section 9 we prove that G l -plurisubharmonic functions and distributionally G l -3lurisubharmonic functions are equivalent in a sense made very precise by Theorem 9.2under the hypothesis that G l involves all the variables and is regular. Strict G l -pluri-subharmonicity can also be defined distributionally and is again equivalent to the viscositydefinition (Theorem 9.8). Section 9 concludes with a local-to-global result (of Richbergtype [R]) for C ∞ approximation of strictly G l -plurisubharmonic functions.Some of the technical issues involving the various hypotheses on G l , such as: G lclosed, G l locally surjective onto X , G l having a fibre-preserving neighborhood retract, or G l modeled on a euclidean case G l , are discussed in Appendix A, in conjunction with adiscussion of the concept of a subequation (Definition A.2) in the geometric case.In appendix B we characterize the subequations which are both linear and geometricunder the weak notion of local jet equivalence (Proposition B.4).Finally we note that the extreme case, where G l = G ( p, T X ) is chosen to be the fullgrassmann bundle, is a basic G l -geometry. There are many additional results specific tothis case which are discussed in a separate but companion paper [HL ]. In that paper weuse the classical terminology: p-plurisubharmonicity, p-convexity, etc. G l -Plurisubharmonicity for Smooth Functions. This concept will be developed in stages. We begin with the basic case.
Euclidean Space.
Suppose V is an n -dimensional real inner product space, and fix an integer p , with1 ≤ p ≤ n . Let Sym ( V ) denote the space of symmetric endomorphisms of V . Using theinner product, this space is identified with the space of quadratic forms on V . Let G ( p, V )denote the set of p -dimensional subspaces of V . For W ∈ G ( p, V ), the W - trace of A ,denoted tr W A , is the trace of the restriction A (cid:12)(cid:12) W of A to W .We identify the Grassmannian G ( p, V ) with a subset of Sym ( V ) by identifying asubspace W with orthogonal projection P W onto the subspace W . The natural innerproduct on Sym ( V ) is defined by using the trace, namely h A, B i = tr( AB ). Under thisidentification we have tr W A = h A, P W i (2 . D x u denote the second derivative of a function u at x ∈ V . Definition 2.1.
Suppose that G l is a closed subset of the Grassmannian G ( p, V ).(a) A form A ∈ Sym ( V ) is G l -positive iftr W A ≥ ∀ W ∈ G l . (2 . u defined on an open subset X ⊂ V is said to be G l -plurisubharmonic iftr W D x u ≥ ∀ W ∈ G l and ∀ x ∈ X. (2 . P ( G l ) denote the set of all G l -positive forms A ∈ Sym ( V ), and let PSH ∞ G l ( X )denote the set of all smooth G l -plurisubharmonic function on X . If tr W A > ∈ G l , then A is said to be G l -strict . Similarly, if the inequalities in (2.3) are all strict,then u is said to be strictly G l -plurisubharmonic .Note that: u ∈ PSH ∞ G l ( X ) ⇐⇒ D x u ∈ P ( G l ) ∀ x ∈ X , and u is G l -strict ⇐⇒ D x u ∈ Int P ( G l ) ∀ x ∈ X The next result justifies the terminology. We shall say that a function u is subharmonicon an affine subspace W if ∆ W (cid:0) u (cid:12)(cid:12) W ∩ X (cid:1) ≥ W is the euclidean Laplacian on W .A p -dimensional affine subspace W is called an affine G l -plane if its corresponding vectorsubspace W is a G l -plane. Proposition 2.2.
A function u ∈ C ∞ ( X ) is G l -plurisubharmonic if and only if therestriction u (cid:12)(cid:12) W ∩ X is subharmonic for all affine G l -planes W ⊂ R n . Proof.
This is obvious from Condition (2) since with v = u (cid:12)(cid:12) W ∩ X , we have tr W D u = ∆ W v on W ∩ X . Riemannian Manifolds.
Suppose X is an n -dimensional riemannian manifold. Then the euclidean notionsabove carry over with V = T x X and the ordinary second derivative of a smooth functionreplaced by the riemannian hessian . Now the set G l will be an arbitrary closed subset ofthe Grassmann bundle π : G ( p, T X ) → X . For u ∈ C ∞ ( X ) this is a well defined sectionof the bundle Sym ( T X ) given on tangent vector fields
V, W by(Hess u )( V, W ) =
V W u − ( ∇ V W ) u, (2 . ∇ denotes the Levi-Civita connection. Note that under composition with a smoothfunction ϕ : R → R , Hess ϕ ( u ) = ϕ ′ ( u )Hess u + ϕ ′′ ( u ) ∇ u ◦ ∇ u (2 . Definition 2.1 ′ . A smooth function u on X is said to be G l -plurisubharmonic if Hess x u is G l x -positive (where G l x = G l ∩ π − ( x )) at each point x ∈ X , i.e.,tr W Hess x u ≥ ∀ W ∈ G l x and ∀ x ∈ X. (2 . ′ Again let PSH ∞ G l ( X ) denote the set of all smooth G l -plurisubharmonic functions on X , and let P ( G l ) denote the subset of Sym ( T X ) with fibres P ( G l x ), the set of G l x -positiveelements in Sym ( T x X ). If the inequalities in (2.3) ′ are all strict at x , then we say that u is strictly G l -plurisubharmonic at x . Exercise 2.1. (Convex Composition Property). If ϕ ∈ C ∞ ( R ) is convex andincreasing, then u ∈ PSH ∞ G l ( X ) ⇒ ϕ ◦ u ∈ PSH ∞ G l ( X ). If, furthermore ϕ is strictlyincreasing and convex, then u strictly G l -psh ⇒ ϕ ◦ u strictly G l -psh.5 xercise 2.2. Show that if u ∈ C ∞ ( X ) is strictly G l -psh at a point x ∈ X , then u isstrictly G l -psh in a neighborhood of x . (See Claim 1 in the proof of Lemma A.3.) Exercise 2.3.
Take X ≡ R and let G l ⊂ G (1 , T X ) = X × G (1 , R ) be defined by setting G l x = G (1 , R ) if x ≥ G l x = ∅ if x <
0. Show that P ( G l ) ⊂ X × Sym ( R ) = R hasfibres R if x < R + = [0 , ∞ ) if x ≥
0. In particular, note that P ( G l ) is not a closedset even though G l is closed. G l -Submanifolds and Restriction. The appropriate geometric objects (in a sense dual to the G l -plurisubharmonic func-tions) are the minimal G l -submanifolds. In the euclidean case this enlarges the family ofaffine G l -planes used in Proposition 2.2. Definition 3.1. If M is a p -dimensional submanifold of X with T x M ∈ G l x for all x ∈ M ,then M is said to be a G l -submanifold. Restriction holds as follows.
THEOREM 3.2.
If a function u ∈ C ∞ ( X ) is G l -plurisubharmonic, then the restriction of u to every minimal G l -submanifold M is subharmonic in the induced riemannian structureon M . Remark 3.3. If G l is determined by a calibration φ , i.e., G l consists of the p -planescalibrated by φ (with the orientation dropped), then G l -submanifolds are automaticallyminimal. Recently, Robles [Ro] has shown that if the calibration is parallel, then thisremains true for any critical set G l corresponding to a non-zero critical value of the cali-bration. Proof.
Suppose M ⊂ X is any p -dimensional submanifold, and let H M denote its meancurvature vector field. Then∆ M (cid:0) u (cid:12)(cid:12) M (cid:1) = tr T M
Hess u − H M u. In particular, if M is minimal, then∆ M (cid:0) u (cid:12)(cid:12) M (cid:1) = tr T M
Hess u. (3 . M is a G l -submanifold, then tr T M
Hess u ≥ Remark 3.4.
If for every point x ∈ X and every p -plane W ∈ G l x , there exists a minimalsubmanifold M with T x M = W , then the converse to Theorem 3.2 is true (use the formula(3.1)). 6 . G l -Convexity and the Core. We will answer four questions concerning the existence of G l -plurisubharmonic func-tions.(1) When does there exist u ∈ PSH ∞ G l ( X ) which is everywhere strict?(2) When does there exist u ∈ PSH ∞ G l ( X ) which is a proper exhaustion for X ?(3) When does there exist u ∈ PSH ∞ G l ( X ) which is both strict and an exhaustion?(4) When does there exist u ∈ PSH ∞ G l ( X ) which is an exhaustion and strict near ∞ ?The answers illustrate some of the flexibility available in constructing G l -plurisubharmonicfunctions.First we characterize those manifolds X which admit a smooth strictly G l -plurisubharmonicfunction. Definition 4.1. (The Core).
The G l -core of X is defined to be the subsetCore G l ( X ) = { x ∈ X : no u ∈ PSH ∞ G l ( X ) is strict at x } . Note that the core is the intersection over u ∈ PSH ∞ G l ( X ) of the closed sets where thegiven u is not strict, and as such is a closed subset of X (see Exercise 2.2). THEOREM 4.2.
The manifold X admits a smooth strictly G l -plurisubharmonic function ⇐⇒ Core G l ( X ) = ∅ . In fact, there exists a function ψ ∈ PSH ∞ G l ( X ) which is G l -strict ateach point x / ∈ Core G l ( X ) . Proof.
The implication ⇒ is clear from the definition. For the converse choose an exhaus-tion of X by compact subsets K ⊂ K ⊂ · · · . Given any sequence of smooth functions u j ∈ C ∞ ( X ) and numbers ǫ j > , j ≥ P ǫ j < ∞ , if we choose numbers δ j > k w k K,j ≡ sup K X | α |≤ j | D α u j | < ǫ j . satisfy δ j k u j k K j ,j ≤ ǫ j , then u = P j δ j u j converges in the C ∞ -topology to u ∈ C ∞ ( X ).If v is G l -strict at a point x , then v is G l -strict in a neighborhood of x (Exercise 2.2).Therefore, if K is a compact set disjoint from Core G l ( X ), then we can find v ∈ PSH ∞ G l ( X )which is G l -strict at each point of K . Hence, we may choose u j ∈ PSH ∞ G l ( X ) with u j strictat each point of K j of distance ≥ /j from Core G l ( X ). Take ψ ≡ P δ j u j as above. Remark.
Essentially the same argument proves that there exists ψ ∈ PSH ∞ G l ( X ) suchthat tr W Hess ψ > G l -planes W which do not lie in the tangential core (see [HL ]). Definition 4.3. (The G l -Convex Hull). Given a subset K ⊂ X , the G l -convex hullof K is the set b K = { x ∈ X : u ( x ) ≤ sup K u ∀ u ∈ PSH ∞ G l ( X ) } . bb K = b K and that b K is closed. THEOREM 4.4. (G l -Convexity and Exhaustion). The following three conditionsare equivalent.(1) If K ⊂⊂ X , then b K ⊂⊂ X .(2) X admits a smooth G l -plurisubharmonic proper exhaustion function u .(3) For some neighborhood of ∞ , X − K with K compact,there exists u ∈ PSH ∞ G l ( X − K ) with lim x →∞ u ( x ) = + ∞ . Condition (3) is a weakening of condition (2) to a local condition at ∞ in the one-pointcompactification X = X ∪ {∞} . Definition 4.5.
We say that X is G l -convex if one of the equivalent condition inTheorem 4.4 holds.The implication (3) ⇒ (2) is immediate from the next (stronger) result. Here K is acompact subset of X . Lemma 4.6.
Given v ∈ PSH ∞ G l ( X − K ) with lim x →∞ v ( x ) = + ∞ , there exists u ∈ PSH ∞ G l ( X ) such that u = v in a neighborhood of ∞ . Proof.
For c sufficiently large, v is smooth and G l -plurisubharmonic outside the compactset { x ∈ X : v ( x ) ≤ c − } . 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 Exercise 2.1, thecomposition ϕ ◦ v is smooth and G l -plurisubharmonic on all of X . Moreover, u = v outsidethe compact set { x ∈ X : v ( x ) ≤ c + 1 } . Proof that (2) ⇒ (1). If K is compact, then c = sup K u < ∞ , and b K is contained inthe compact set { u ≤ c } .The implication (1) ⇒ (2) is a construction using the next lemma. Lemma 4.7.
Suppose K ⊂ X is compact. If x / ∈ b K , then there exists u ∈ PSH ∞ G l ( X ) satisfying:(a) u ≡ on a neighborhood of K ,(b) u ( x ) > , and(c) u is strict at x if x / ∈ Core G l ( X ) . Proof.
Suppose x / ∈ b K . Then there exists v ∈ PSH ∞ G l ( X ) with sup K v < < v ( x ). Pick ϕ ∈ C ∞ ( R ) with ϕ ≡ −∞ ,
0] and with ϕ > , ∞ ). Then u = ϕ ◦ v satisfies the required conditions. Furthermore, assume h ∈ PSH ∞ G l ( X ) is strictat x . Then take v = v + ǫh . For small enough ǫ , sup K v < < v ( x ). If ϕ is also strictlyincreasing on (0 , ∞ ), then u = ϕ ◦ v is strict at x . Proof that (1) ⇒ (2). A G l -plurisubharmonic proper exhaustion function on X isconstructed as follows. Choose an exhaustion of X by compact G l -convex subsets K ⊂ K ⊂ K ⊂ · · · with K m ⊂ K m +1 for all m . By Lemma 4.7 and the compactness8f K m +2 − K m +1 , there exists a G l -plurisubharmonic function f m ≥ X with f m identically zero on a neighborhood of K m and f m > K m +2 − K m +1 . By re-scalingwe may assume f m > m on K m +2 − K m +1 . The locally finite sum f = P ∞ m =1 f m satisfies(2). Next we characterize the existence of a strict exhaustion function. THEOREM 4.8. (Strict G l -Convexity). The following conditions are equivalent:(1)
Core G l ( X ) = ∅ , and if K ⊂⊂ X , then b K ⊂⊂ X ,(2) X admits a smooth proper exhaustion function which is strictly G l -plurisubharmonic. Proof that (1) ⇒ (2). Since Core G l ( X ) = ∅ , there exists a strictly G l -plurisubharmonicfunction v by Proposition 4.2. If u is a G l -plurisubharmonic exhaustion function given byTheorem 4.4, then u + e v is a strict exhaustion. Definition 4.9.
We say that X is strictly G l -convex if one of the equivalent conditionsof Theorem 4.8 holds. Corollary 4.10.
Suppose that
Core G l ( X ) = ∅ . If X is G l -convex , then X is strictly G l -convex . THEOREM 4.11. (Strict G l -Convexity at Infinity). The following conditions areequivalent:(1)
Core G l ( X ) is compact, and if K ⊂⊂ X , then b K ⊂⊂ X ,(2) X admits u ∈ PSH G l ( X ) with lim x →∞ u ( x ) = ∞ and u strict outside acompact subset.(3) Core G l ( X ) is compact, and X admits u ∈ PSH G l ( X − K ) , for some compactsubset K , with lim x →∞ u ( x ) = ∞ . Proof that (3) ⇒ (2). Apply Lemma 4.6.
Proof that (2) ⇒ (1). (Straightforward) Proof that (1) ⇒ (3). Core G l ( X ) ≡ K is compact ⇒ Core G l ( X − K ) = ∅ . Definition 4.12.
We say that X is strictly G l -convex at infinity if one of the equivalentcondition in Theorem 4.11 holds.Some of the previous results can be summarized as follows. Corollary 4.13.
Suppose Core G l ( X ) = ∅ . Then the following are equivalent.(1) X is G l -convex.(2) X is strictly G l -convex.(3) X is strictly G l -convex at infinity. Proof.
Use Theorems 4.4 and 4.11. 9 roposition 4.14.
Suppose ( M, ∂M ) is a compact connected G l -submanifold-with-boundary in X . If M is minimal (stationary), then(1) If ∂M = ∅ , then M ⊂ Core G l ( X ) .(2) If ∂M = ∅ , then M ⊂ d ∂M . Proof.
Since the restriction of any u ∈ PSH ∞ G l ( X ) to M is subharmonic on M , themaximum principle applies to u (cid:12)(cid:12) M .This proposition provides an analogue of the support Lemma 3.2 in [HL ]:If M is a minimal G l submanifold , then M ⊂ d ∂M ∪ Core G l ( X ) . The existence question for strictly G l -convex manifolds has two sides. We brieflymention these results from both [HL ] and [HL ]. Definition 4.15. (G l -Free). A subspace V ⊂ T X is said to be G l -free if there are no G l -planes contained in V . The maximal dimension of such a free subspace, taken overall points x ∈ X , is called the free dimension of G l and is denoted freedim( G l ). Asubmanifold M of X is G l -free if T x M is G l -free for each x ∈ M .Strict G l -convexity of X imposes conditions on the topology of X . THEOREM 4.16.
A strictly G l -convex manifold has the homotopy type of a CW complexof dimension ≤ freedim ( G l ) . The free dimension of G is computed in many examples in [HL ] and summarized in[HL ].On the other hand, the existence of many strictly G l -convex manifolds is guaranteedby another result (see Theorem 6.6 in [HL ]). THEOREM 4.17.
Suppose M is a G l -free submanifold of X . Then M has a fundamentalneighborhood system in X consisting of strictly G l -convex manifolds, each of which has M as a deformation retract.
5. Boundary Convexity
Suppose that Ω ⊂ X is an open connected set with smooth non-empty boundary ∂ Ωcontained in an oriented riemannian manifold. Fix a closed subset G l ⊂ G ( p, T X ). Definition 5.5. A p -plane W ∈ G l x at x ∈ ∂ Ω is called a tangential G l -plane at x if W ⊂ T x ( ∂ Ω).Denote by II = II ∂ Ω the second fundamental form of the boundary with respect tothe inward pointing normal n . This is a symmetric bilinear form on each tangent space T x ( ∂ Ω) defined by II ( v, w ) = −h∇ v n, w i = h n, ∇ v W i where W is any vector field tangent to ∂ Ω with W x = w .10 efinition 5.2. The boundary ∂ Ω is G l -convex at a point x if tr W II x ≥ G l -planes W at x . If this inequality is strict, then we say that ∂ Ω is strictlyG l -convex at x . Definition 5.3. (Local defining functions).
Suppose ρ is a smooth function on aneighborhood B of a point x ∈ ∂ Ω with ∂ Ω ∩ B = { ρ = 0 } and Ω ∩ B = { ρ < } . If dρ isnon-zero on ∂ Ω ∩ B , then ρ is called a local defining function for ∂ Ω. Lemma 5.4. If ρ is a local defining function for ∂ Ω , then for all x ∈ ∂ Ω ∩ B , Hess x ρ (cid:12)(cid:12) T x ( ∂ Ω) = |∇ ρ ( x ) | II x Proof.
Suppose that e is a vector field on B tangent to ∂ Ω along ∂ Ω, and note that II ( e, e ) = h n, ∇ e e i = − |∇ ρ | h∇ ρ, ∇ e e i and −h∇ ρ, ∇ e e i = − ( ∇ e e )( ρ ) = e ( eρ ) − ( ∇ e e )( ρ ) =(Hess ρ )( e, e ). Corollary 5.5.
The boundary ∂ Ω is G l -convex at a point x if and only if tr W Hess ρ ≥ G l − planes W tangent to ∂ Ω at x (5 . where ρ is a local defining function for ∂ Ω . In particular the condition (5.1) is independentof the choice of local defining function ρ . Moreover, the boundary is strictly G l -convex ata point x if and only if the inequalities in(5.1) are all strict, again with independence ofthe choice of ρ . Remark 5.6. If ∂ Ω is G l -free at a point x ∈ ∂ Ω (see Definition 4.15), then ∂ Ω is auto-matically strictly G l -convex at x since there are no tangential G l -planes W to consider. Forexample, in the extreme case p = n (the Laplacian subequation) all boundaries ∂ Ω arestrict at each point since all hyperplanes in T x X are G l -free. THEOREM 5.7.
Suppose that ∂ Ω is strictly G l -convex. Then there exists a global G l -plurisubharmonic defining function ρ ∈ C ∞ (Ω) which is strict on a collar {− ǫ ≤ ρ ≤ } .If Core (Ω) = ∅ , then ρ can be chosen to be strict on all of Ω . Corollary 5.8. If ∂ Ω is strictly G l -convex, then Ω is strictly G l -convex at ∞ ; and ifCore (Ω) = ∅ , then Ω is strictly G l -convex. Proof of Corollary.
Suppose that ρ ∈ C ∞ (Ω) is a defining function for ∂ Ω. Then − log( − ρ ) is an exhaustion funtion for Ω. Since the function ψ : ( −∞ , → ( −∞ , ∞ )defined by ψ ( t ) = − log( − t )is strictly convex and increasing, − log( − ρ ) is strictly G l − plurisubharmonic at points in Ωwhere ρ is strictly G l − plurisubharmonic . (5 . Proof of Theorem.
Start with an arbitrary defining function ρ ∈ C ∞ (Ω) for ∂ Ω. Set e ρ ≡ ρ + λ ρ with λ >
0. Then at points in ∂ ΩHess e ρ = (1 + λρ )Hess ρ + λ ∇ ρ ◦ ∇ ρ = Hess ρ + λ ∇ ρ ◦ ∇ ρ. (5 . λ sufficiently large , e ρ = ρ + λ ρ is strictly G l − plurisubharmonicat every boundary point x ∈ ∂ Ω . (5 . e ρ is strictly G l -plurisubharmonic in a neighborhood of ∂ Ω in X , andhence on some collar {− ǫ ≤ e ρ ≤ } with ǫ >
0. Choose ψ ( t ) convex and increasing with ψ ( t ) ≡ − ǫ if t ≤ − ǫ , and ψ ( t ) = t if t ≥ − ǫ . Then ψ ( e ρ ) is G l -plurisubharmonic on Ω andequal to e ρ on the collar {− ǫ ≤ e ρ ≤ } , thereby providing the required defining function. IfCore(Ω) is empty, then add the global strictly G l -plurisubharmonic function, provided byTheorem 4.2, to ψ ( e ρ ).It remains to prove (5.4). Each p -plane V ∈ G ( p, T x X ) can be put in canonical formwith respect to T x ∂ Ω. Let n denote a unit normal to T x ∂ Ω in T x X . Choose an orthonormalbasis e , ..., e p for V such that e , ..., e p is an orthonormal basis for V ∩ ( T x ∂ Ω). Then e = cos θ V n + sin θ V e defines an angle θ V mod π and a unit vector e ∈ T x ∂ Ω. Now by(5.3) we have tr V Hess e ρ = tr V Hess ρ + λ cos θ V |∇ ρ | . (5 . | cos θ V | < δ defines a fundamental neighborhood system for G ( p, T ∂ Ω)as a subset of the bundle G ( p, T X ) (cid:12)(cid:12) ∂ Ω . Intersecting with G l (cid:12)(cid:12) ∂ Ω we see that G l ∩ G ( p, T ∂ Ω)has a fundamental neighborhood system in G l (cid:12)(cid:12) ∂ Ω given by N δ ≡ { V ∈ G l x : x ∈ ∂ Ω and | cos θ V | < δ } . Since ∂ Ω is strictly G l -convex, there exists η > W Hess ρ ≥ η for all W ∈ G l ∩ G ( p, T ∂ Ω). Hence for δ small, tr V Hess ρ ≥ η for all V ∈ N δ . Choose a lower bound − M for tr V Hess ρ over all V ∈ G l (cid:12)(cid:12) ∂ Ω .Assume V ∈ G l x , x ∈ ∂ Ω. For | cos θ V | < δ , tr V Hess e ρ ≥ η + λ cos θ V |∇ ρ | ≥ η . For | cos θ V | ≥ δ , tr V Hess e ρ ≥ − M + λδ |∇ ρ | which is ≥ η if λ is chosen large. This proves(5.4). Remark 5.9.
Simple examples show that strict G l -convexity of ∂ Ω does not imply thatevery defining function ρ for ∂ Ω is strictly G l -plurisubharmonic at points of ∂ Ω. However,the exhaustion − log( − ρ ) is always strictly G l -plurisubharmonic on a small enough collarof ∂ Ω. For the proof of this, compute Hess( − log( − ρ )) and mimick the proof of Theorem5.7 on the hypersurfaces { ρ = ǫ } (see the proof of Theorem 5.6 in [HL ]). Remark 5.10. (Signed Distance).
Recall that a defining function ρ for Ω satisfies |∇ ρ | ≡ ∂ Ω if and only if ρ is the signed distance to ∂ Ω ( < > ρ with |∇ ρ | ≡ ρ = (cid:18) II (cid:19) (5 . II denotes the second fundamental form of the hypersurface H = { ρ = ρ ( x ) } withrespect to the normal n = −∇ ρ and the blocking in (5.6) is with respect to the splitting T x X = N x H ⊕ T x H . For example let ρ ( x ) = k x k ≡ r in R n . Then direct calculationshows that Hess ρ = r ( I − ˆ x ◦ ˆ x ) where ˆ x = x/r . Moreover,Hess( ρ + λρ ) = (cid:18) λ II (cid:19) (5 . δ = − ρ ≥
0, the actual distance to ∂ Ω inΩ, we have Hess( − log δ ) = 1 δ (cid:18) δ II (cid:19) (5 . ρ . Namely, with δ ( x ) ≡ dist( x, ∂ Ω) we havethat ∂ Ω strictly G l − convex ⇒ − log δ is strictly G l − psh in a collar . (5 . Remark 5.11. (G l -Parallel). If G l is parallel as a subset of G ( p, T X ) ⊂ Sym ( T X ),then a weakened form of the converse to (5.9) is true. Namely, If − log δ is G l -plurisubharmonic in a collar, then ∂ Ω is G l -convex at each point. Proof. If ∂ Ω is not G l -convex at x ∈ ∂ Ω, then with ρ ≡ − δ , tr W Hess x ρ < W ∈ G l x tangential to ∂ Ω at x . let γ denote the geodesic segment in Ω which emanatesorthogonally from ∂ Ω at x . Since δ is the distance function to ∂ Ω, γ is an integral curveof ∇ δ . Let W y denote the parallel translate of W along γ to y . Then W y ∈ G l y and( ∇ δ ) y ⊥ W y . Therefore by (5.8), tr W y Hess y ( − log δ ) = δ tr W y Hess y ( ρ ) < y sufficientlyclose to x . Hence − log δ is not G l -plurisubharmonic near ∂ Ω. Local Convexity of a Domain Ω ⊂ X For simplicity assume that Core G l ( X ) is empty. Then for each open subset Y ⊂ X thethree notions of convexity, namely G l -convexity, strict G l -convexity, and strict G l -convexityat infinity, are all equivalent. Definition 5.12.
A domain Ω ⊂ X is locally G l -convex if each point x ∈ ∂ Ω has aneighborhood U in X such that U ∩ Ω is G l -convex.Small balls are G l -convex and the intersection of two G l -convex domains is again G l -convex. Therefore:If Ω is G l − convex , then Ω is locally G l − convex . (5 . X, G l doesΩ locally G l − convex ⇒ Ω is G l − convex? (5 . Example 5.13. (Horizontal convexity in R ). Take G l = { R × { }} ⊂ G (1 , R ) asingleton consisting of the x -axis. A domain is G l -convex if and only if all of its horizontalslices are connected. Choose Ω ⊂⊂ R with the property that ∂ Ω contains the interval[ − ,
1] on the x -axis, the lower half of the circle of radius 3 about the origin, and the points( − , , (2 , G l -convex but not globally G l -convex. Inaddition, the boundary of Ω can be made G l -convex.By contrast, one of the main results of [HL ] is the solution to the Levi Problem ineuclidean space in the extreme case G l = G ( p, R n ).13 . Upper Semi-Continuous G l -Plurisubharmonic Functions. Let X be a riemannian manifold, and assume that G l ⊂ G ( p, T X ) is a closed subset.Denote by USC( X ) the space of upper semi-continuous [ −∞ , ∞ )-valued functions on X .By a test function for u ∈ USC( X ) at a point x we mean a C -function ϕ , defined near x , such that u ≤ ϕ near x and u ( x ) = ϕ ( x ). Definition 6.1.
A function u ∈ USC( X ) is G l -plurisubharmonic if for each x ∈ X andeach test function ϕ for u at x , the riemannian hessian Hess x ϕ at x satisfiestr W Hess x ϕ ≥ ∀ W ∈ G l x i.e., Hess x ϕ ∈ P ( G l x ). The space of these functions is denoted by PSH G l ( X ).This definition is an extension of Definition 2.1 ′ because of the following. Lemma 6.2.
Suppose u ∈ C ( X ) . Then for a point x ∈ X , the following are equivalent: tr W Hess x ϕ ≥ ∀ W ∈ G l x and all test functions ϕ for u at x, (6 . W Hess x u ≥ ∀ W ∈ G l x , (6 . Proof.
Note that (6.1) ⇒ (6.2) because we can take ϕ = u in (6.1). Assume (6.2) and that ϕ is a test function for u at x . Then ψ ≡ ϕ − u ≥ x and vanishes at x . Hence x is acritical point for ψ , and the second derivative or hessian of ψ is a well defined non-negativeelement of Sym ( T x X ), independent of any metric. In particular, tr W Hess x ψ ≥ W ∈ G ( p, T x X ). Since Hess x ϕ = Hess x u + Hess x ψ , taking the W -trace with W ∈ G l x , wesee that (6.2) ⇒ (6.1). Remark 6.3. (Positivity).
Let P x ⊂ Sym ( T x X ) denote the subset of non-negativeelements. Replacing P ( G l ) ⊂ Sym ( T X ) with a general closed subset F ⊂ Sym ( T X ), theabove (standard) proof shows that (6.2) implies (6.1), i.e., Hess x u ∈ F x ⇒ Hess x ϕ ∈ F x ,provided that F satisfies the positivity condition : F x + P x ⊂ F x for all x ∈ X. ( P )There are several equivalent ways of stating the condition (6.1). We record one thatis particularly useful, and refer the reader to Appendix A in [HL ] for the proof as well asthe statements of the other conditions. Lemma 6.4.
Suppose u ∈ USC( X ) . Then u / ∈ PSH G l ( X ) if and only if ∃ x ∈ X, α > ,and a smooth function ϕ defined near x satisfying: u − ϕ ≤ − α | x − x | near x u − ϕ = 0 at x but with tr W Hess x ϕ < for some W ∈ G l x . lementary Properties Even though G l ⊂ Sym ( T X ) is closed, the subset P ( G l ) ⊂ Sym ( T X ) of G l -positiveelements may not be closed (see Exercise 2.3). However, by Proposition A.6 below, P ( G l )is closed if and only if π (cid:12)(cid:12) G l is a local surjection. We make this assumption unless thecontrary is stated.The following basic facts can be found for example in [HL , Theorem 2.6]. In factthey hold with P ( G l ) replaced by any subequation (see Definition A.2). THEOREM 6.5. (a) (Maximum Property) If u, v ∈ PSH G l ( X ), then w = max { u, v } ∈ PSH G l ( X ).(b) (Coherence Property) If u ∈ PSH G l ( X ) is twice differentiable at x ∈ X , then Hess x u is G l -positive.(c) (Decreasing Sequence Property) If { u j } is a decreasing ( u j ≥ u j +1 ) sequence of func-tions with all u j ∈ PSH G l ( X ), then the limit u = lim j →∞ u j ∈ PSH G l ( X ).(d) (Uniform Limit Property) Suppose { u j } ⊂ PSH G l ( X ) is a sequence which convergesto u uniformly on compact subsets to X , then u ∈ PSH G l ( X ).(e) (Families Locally Bounded Above) Suppose F ⊂
PSH G l ( X ) is a family of functionswhich are locally uniformly bounded above. Then the upper semicontinuous regular-ization v ∗ of the upper envelope v ( x ) = sup u ∈F u ( x )belongs to PSH G l ( X ). Example 6.6.
The following examples show that Properties (c), (d) and (e) require thatthe set P ( G l ) be closed. Let X = R and G l x = { T x R } ∈ G (1 , T X ) if x ≥ G l x = ∅ for x <
0. Note that G l is a closed set. Then P ( G l x ) = Sym ( T x X ) ∼ = R for x < P ( G l x ) = { A ∈ Sym ( T x X ) : A ≥ } for x ≥
0. Note that P ( G l ) is not closed in R × R .This subequation is simply the requirement that u ′′ ( x ) ≥ x ≥ . Fix a constant a > u ( x ) = (cid:26) x ≥ ,x ( a − x ) if x ≤ . This function fails to be G l -plurisubharmonic at 0. To see this note that ϕ ( x ) = x ( a − x )is a test function for u at 0 and ϕ ′′ (0) < δ > v δ ( x ) = u ( x + δ ) + δ . Note that graph( v δ ) = graph( u ) + ( − δ, δ ).Then each v δ is G l -plurisubharmonic and v δ ↓ u as δ →
0. Hence condition (c) fails.Now for each ǫ >
0, define u ǫ ≡ min { u, − ǫ } . Then u ǫ is G l -plurisubharmonic for all ǫ and u ǫ ↑ u as ǫ →
0. Hence conditions (d) and (e) also fail.15 estriction
Throughout this subsection we assume that G l ⊂ G ( p, T X ) is a closed set admitting asmooth neighborhood retraction preserving the fibres of the projection π : G ( p, T X ) → X .The terminology G l -plurisubharmonic for u ∈ USC( X ) is justified by the next result, whichextends Theorem 3.2. THEOREM 6.7. If u ∈ PSH G l ( X ) , then for every minimal G l -submanifold M , therestriction u (cid:12)(cid:12) M is ∆ -subharmonic where ∆ is the Laplace-Beltrami operator in the inducedriemannian metric on M . This result can be extended to submanifolds M of dimension larger that p . Let G l M ≡ { W ∈ G ( p, T M ) : W ∈ G l } denote the set of tangential G l -planes to M . Thisset G l M defines a notion of G l M -plurisubharmonicity for functions w ∈ USC( M ). Definition 6.8.
We say that G l is regular if at every point x ∈ X , each element W ∈ G l x has a local smooth extension to a section W ( x ) of G l . Definition 6.9.
A submanifold M of X is G l -flat if the second fundamental form B of M satisfies tr (cid:0) B (cid:12)(cid:12) W (cid:1) = 0 for all tangential G l planes W ∈ G l M (6 . THEOREM 6.10.
Suppose M is a G l -flat submanifold of X and that the subset G l M ⊂ G ( p, T M ) is regular on M . If u ∈ PSH G l ( X ) , then u (cid:12)(cid:12) M ∈ PSH G l M ( M ) . See Section 8 of [HL ] for a more complete discussion, including Example 8.4, whichshows that G l M being regular is necessary in Theorem 6.10. The proof uses Lemma 8.3 in[HL ] which is stated in this paper as Proposition 8.4 below. G l -Harmonic Functions and the Dirichlet Problem. In this section we discuss the Dirichlet problem for extremal or G l -harmonic functions.These are natural generalizations of solutions of the classical homogeneous Monge-Amp`ereproblem, in both the real and complex cases (and constitute a very special case of thegeneral F -harmonic functions treated in [HL ]). To do this we must introduce the Dirichletdual . Dually G l -Plurisubharmonic Functions We first define the
Dirichlet dual of the subset F ≡ P ( G l ) ⊂ Sym ( T X ), to be thesubset e F ≡ g P ( G l ) ⊂ Sym ( T X ) whose fibres are given by f F x = − ( ∼ Int F x ) = ∼ ( − Int F x ) . (7 . A ∈ Int F x ⇐⇒ tr W A > W ∈ G l x , (7 . A ∈ f F x ⇐⇒ tr W A ≥ W ∈ G l x , (7 . efinition 7.1. A smooth function u on X is said to be dually G l -plurisubharmonic if at each point x ∈ X ∃ W ∈ G l x with tr W Hess x u ≥ , or equivalently Hess x u ∈ g P ( G l ) . More generally a function u ∈ USC( X ) is dually G l -plurisubharmonic if for each point x ∈ X and each test function ϕ for u at x , ∃ W ∈ G l x with tr W Hess x ϕ ≥ , or equivalently Hess x ϕ ∈ g P ( G l ) . The set of all such functions is denoted g PSH G l ( X ).First note that g P ( G l ) satisfies the positivity condition (P), so that as noted in Remark6.3, if a smooth function u satisfies Hess x u ∈ g P ( G l ), then for each test function ϕ for u at x ,we have Hess x ϕ ∈ g P ( G l ), making the second definition an extension of the first definition.Second, assuming that π (cid:12)(cid:12) G l is a local surjection as in Definition A.5, it then follows thatnot only P ( G l ), but also g P ( G l ) is closed. As a consequence,the set g PSH G l ( X ) satisfies all of the properties given in Theorem 6 . . (7 . g P ( G l ) is a subequation (Definition A.2).By Theorem 4.2 if Core G l ( X ) = ∅ , then X admits a smooth function ψ whichis strictly G l -plurisubharmonic at each point. Of course, P ( G l ) ⊂ g P ( G l ), so that thedually G l -plurisubharmonic functions on X constitute a much larger class than the G l -plurisubharmonic functions. Again we assume that π (cid:12)(cid:12) G l is a local surjection. THEOREM 7.2. (The Maximum Principle for Dually G l -PlurisubharmonicFunctions). Suppose
Core G l ( X ) = ∅ . Then for each compact subset K ⊂ X and each u ∈ g PSH G l ( K ) ≡ USC( X ) ∩ g PSH G l (Int K ) we have: sup K u ≤ sup ∂K u. The proof is classical and completely elementary. Moreover, one can easily see thatthis maximum principle is equivalent to the special case of comparison (Theorem 7.7 below)where u is smooth. Proof.
Suppose it fails. Then there exist a compact set K , a function u ∈ g PSH G l ( K ) anda point ¯ x ∈ Int K with u (¯ x ) > sup ∂K u . Let ψ be a smooth strictly G l -psh function on X . Then for ǫ > u + ǫψ will also have a maximum atsome point x ∈ Int K . Thus − ǫψ is a test function for u at x , and therefore Hess x ( − ǫψ ) ∈ g P x ( G l ) = − ( ∼ Int P x ( G l )), i.e., Hess x ( ψ ) / ∈ Int P x ( G l ) contradicting the strictness of ψ at x .17he Convex-Increasing Composition Property in Exercise 2.1 not only extends to theupper semi-continuous case, but also to the much larger class of dually G l -plurisubharmonicfunctions. Lemma 7.3. (Composition Property).
Suppose ϕ : R → R is both convex andincreasing (i.e., non-decreasing). Then u ∈ g PSH G l ( X ) ⇒ ϕ ◦ u ∈ g PSH G l ( X ) ( a ) If ϕ is also strictly increasing, then in addition to (a) we have that u is G l strict ⇒ ϕ ◦ u is G l strict ( b ) where we refer ahead to Definition 7.7 for the notion of strictness. Proof.
We can assume that ϕ is smooth since it can be approximated by a decreasingsequence ϕ ǫ via convolution. Observe now that: ψ is a test function for u at x ⇐⇒ ϕ ◦ ψ is a test function for ϕ ◦ u at x . This reduces the proof to the case where ϕ and u are both smooth, and formula (2.5)applies with both coefficients ϕ ′ ( u ( x )) and ϕ ′′ ( u ( x )) ≥ G l -Harmonics To understand the next definition note that ∂ P ( G l ) = P ( G l ) ∩ ( − g P ( G l )) (7 . Definition 7.4.
A function u on X is said to be G l -harmonic if u ∈ PSH G l ( X ) and − u ∈ g PSH G l ( X ) . By (7.5) we see that a C -function u on X is G l -harmonic if and only ifHess x u ∈ ∂ P ( G l x ) for all x ∈ X. In order to solve the Dirichlet Problem for G l -harmonic functions on domains Ω ⊂ X ,we restrict G l ⊂ G ( p, T X ) to be modeled on a “constant coefficient” case G l ⊂ G ( p, R n ). Definition 7.5.
A closed subset G l ⊂ G ( p, T X ) is locally trivial with fibre G l ⊂ G ( p, R n ), if in a neighborhood each point x ∈ X there exists a local tangent frame fieldso that under the associated trivialization φ : G ( p, T U ) ∼ = −−−→ U × G ( p, R n ) we have φ : G l (cid:12)(cid:12) U ∼ = −−−→ U × G l . This can be formulated somewhat differently. Let Aut( G l ) = { g ∈ GL n : g ( G l ) = G l } .Then given a closed subset G l ⊂ G ( p, T X ) which is locally trivial with fibre G l , the localtangent frame fields in Definition 7.5 provide X with a topological Aut( G l )-structure (see18 ]). Conversely, if X admits a topological Aut( G l )-structure, then the euclideanmodel G l ⊂ G ( p, R n ) determines a canonical closed subset G l ⊂ G ( p, T X ) which is locallytrivial with fibre G l . In other words, a euclidean model can be transplanted to anymanifold with a topological Aut( G l )-structure (again see § ]).In the language of [HL , § G l is locally trivial with fibre G l ” means that thesubequation P ( G l ) is locally jet equivalent to the constant coefficient subequation P ( G l ).In the next two theorems X is a riemannian manifold and G l ⊂ G ( p, T X ) is a closed,locally trivial set with non-empty fibre. THEOREM 7.6. (The Dirichlet Problem).
Suppose that Ω ⊂⊂ X is a domain witha smooth, strictly G l -convex boundary ∂ Ω and Core G l (Ω) = ∅ . Then the Dirichlet problemfor G l -harmonic functions is uniquely solvable on Ω . That is, for each ϕ ∈ C ( ∂ Ω) , thereexists a unique G l -harmonic function u ∈ C (Ω) such that(i) u (cid:12)(cid:12) Ω is G l -harmonic, and(ii) u (cid:12)(cid:12) ∂ Ω = ϕ. This is the special case Theorems 16.1 of Theorem 13.1 in [HL ]. There are manyinteresting examples. See [HL ] for a long list.Boundary convexity is not required for uniqueness, only an empty core for X . Asusual uniqueness is immediate from comparison. THEOREM 7.7. (Comparison).
Suppose that Core G l ( X ) = ∅ and K ⊂ X is compact.If u ∈ PSH G l ( K ) and v ∈ g PSH G l ( K ) , then the zero maximum principle holds, that is, u + v ≤ on ∂K ⇒ u + v ≤ on K. (ZMP) Outline of proof.
By definition u, v ∈ USC( K ) and on the interior Int K , u is G l -plurisubharmonic and v is dually G l -plurisubharmonic. The appropriate notion of strict plurisubharmonicity for general upper semi-continuous functions plays a crucial role, andwill be discussed below after outlining its importance. If (ZMP) holds for all compact K ⊂ X under the additional assumption that u is G l -strict, we say that weak comparisonholds for G l on X . This weakened version of comparison has one big advantage, namelythat local implies global (Theorem 8.3 in [HL ]). The proof of completed by showing twothings. First, Weak comparison is true locally . (7 . ].Second, strict approximation holds. That is, since Core G l ( X ) = ∅ , X supports a C strictly G l -plurisubharmonic function ψ , andIf u is G l − plurisubharmonic , then u + ǫψ is strictly G l − plurisubharmonic , for each ǫ > . (7 . trictnessDefinition 7.8. A function u ∈ USC( X ) is strictly G l -plurisubharmonic if each pointin X has a neighborhood U along with a constant c > x ∈ U and each test function ϕ for u at x tr W Hess x ϕ ≥ c for all W ∈ G l x . (7 . , Def. 7.4,]one must compare (7.8) with distance in Sym ( T x X ). For this first note that for W ∈ G ( p, T x X ) the (signed) distance of a point A ∈ Sym ( T x X ) to the boundary of the positivehalf-space defined by the unit normal p P W is simply h A, p P W i . Consequently, the distancefrom A ∈ P ( G l x ) to ∼ P ( G l x ) is given bydist( A, ∼ P ( G l x )) = inf W ∈ G l x h A, p P W i = inf W ∈ G l x p tr W A. (7 . c > c -strictness is a subequation. Therefore, all the propertiesin Theorem 6.5 hold for c -strict G l -plurisubharmonic functions. Moreover, as noted inLemma 7.3, if ϕ is convex and strictly increasing, the composition property holds. Finally,strictness is “stable”. Lemma 7.9. ( C ∞ -Stability Property). Suppose u is strictly G l -plurisubharmonic and ψ ∈ C ∞ ( X ) with compact support. Then u + ǫψ is strictly G l -plurisubharmonic for all ǫ sufficiently small. Proof.
This is Corollary 7.6 in [HL ].
8. Geometric Subequations Involving all the Variables.
This is a concept which distinguishes, for example, the full Laplacian on R n , whichinvolves all the variables, from the p th partial Laplacian ∆ p , which does not. We shall firsttreat the euclidean case (see Section 2 of [HL ]). The results will then be carried over toa general riemannian manifold X .Fix a finite dimensional inner product space V and suppose G l ⊂ G ( p, V ) is a closedsubset of the grassmannian. Let Span G l denote the span in Sym ( V ) of the elements P W with W ∈ G l , and let P + ( G l ) denote the convex cone on G l with vertex at the origin inSym ( V ). Examples show that Span G l is often a proper vector subspace of Sym ( V ),in which case P + ( G l ) will have no interior in Sym ( V ). However, considered as a subsetof the vector space Span G l , the interior of P + ( G l ) has closure equal to P + ( G l ). Wedefine Int P + ( G l ) to be the interior of P + ( G l ) in Span G l (not in Sym ( V )). In particular,Int P + ( G l ) is never empty, and P + ( G l ) = Int P + ( G l ).By Definition 2.1, P ( G l ) = { B ∈ Sym ( V ) : h B, P W i ≥ W ∈ G l } . Hence, P ( G l ) ⊂ H ( A ) for each closed half-space H ( A ) ≡ { B ∈ Sym ( V ) : h A, B i ≥ } determinedby a non-zero A ∈ P + ( G l ). This proves that P ( G l ) = \ A ∈P + ( G l ) H ( A ) , P ( G l ) is the “polar” of P + ( G l ). (Therefore, by the Hahn-Banach/Bipolar Theorem P + ( G l ) is the polar of P ( G l ).)Since P + ( G l ) = Int P + ( G l ), this intersection can be taken over the smaller set of A ∈ Int P + ( G l ). That is, P ( G l ) = \ A ∈ Int P + ( G l ) H ( A ) . (8 . G l insuresthat such A are positive definite, i.e., the linear operators h A, D u i are uniformly elliptic.The linear operator ∆ A u ≡ h A, D u i with A ≥ A -Laplacian . Note that from our set theoretic point of view, the subequation ∆ A ⊂ Sym ( V ) is precisely the closed half-space H ( A ).The following is a restatement of Proposition 2.8 in [HL ] (see also Remark 4.8, page874 of [K]). Lemma 8.1.
The following are equivalent ways of defining the concept that G l involvesall the variables .(1) The only vector v ∈ Sym ( V ) with v ⊥ W for all W ∈ G l is v = 0 .(2) For each unit vector e ∈ V , P e is never orthogonal to Span G l .(3) There does not exist a hyperplane W ⊂ V with G l ⊂ Sym ( W ) ⊂ Sym ( V ) .(4) Int P + ( G l ) ⊂ Int P , i.e., each A ∈ Int P + ( G l ) is positive definite.(5) There exists A ∈ Span G l with A > . In Section 2 of [HL ] such subsets G l were called “elliptic”.We shall apply Lemma 8.1 to the case V = T x X on a riemannian manifold X . Wesay that a closed subset G l ⊂ G ( p, T X ) involves all the variables if each fibre G l x ⊂ G ( p, T x X ) involves all the variables in the vector space V ≡ T x X . For any smooth section A ( x ) ≥ ( T X ) the linear operator∆ A u ≡ h A ( x ) , Hess x u i will again be referred to as the A -Laplacian .Recall from Definition 6.8 that G l is regular if each element W ∈ G l x can be locallyextended to a smooth section W ( y ) of G l . This immediately implies that each element A ∈ P + ( G l x ) can be locally extended to a smooth section A ( y ) with A ( y ) ∈ P + ( G l y ),(since A = P k t k W k for t k > W k ∈ G l x ). Furthermore, if A ( x ) >
0, then A ( y ) > y near x . This proves the following. Lemma 8.2.
Suppose G l ⊂ G ( p, T X ) is a closed subset involving all the variables andthat G l is regular. Then P ( G l x ) = \ H ( A ( x )) for each x ∈ X (8 . ′ where the intersection is taken over all smooth P + ( G l ) -valued section A ( y ) where A ( y ) > for y near x . orollary 8.3. A function u ∈ USC( X ) is G l -plurisubharmonic ⇐⇒ u is ∆ A -subharmonic for each smooth (local) section A of Sym ( T X ) with values in P + ( G l ) and A > . Proof. If A is a section of P + ( G l ), then P + ( G l ) ⊂ ∆ A over a neighborhood U of x , so thateach G l -plurisubharmonic function on U is automatically ∆ A -subharmonic. Conversely, if u is ∆ A -subharmonic for each (local) section A of P + ( G l ) with A >
0, and if ϕ is a testfunction for u at x , then Hess x ϕ ∈ H ( A ( x )), and therefore by (8.1) ′ , Hess x ϕ ∈ P ( G l x ). Note 8.4.
The simple argument just given also shows the following.
Suppose F is asubequation on X which can be written as an intersection of subequations F = T α F α .Then for u ∈ USC( X ) , u is F -subharmonic if and only if u is F α -subharmonic for all α . Corollary 8.3 has many consequences. We mention one.
THEOREM 8.5. (The Strong Maximum Principle for G l -PlurisubharmonicFunctions). Suppose G l ⊂ G ( p, T X ) is regular and involves all the variables. Then forany compact subset K with Int K connected and K = Int K , if u ∈ PSH G l ( K ) has aninterior maximum point, then u (cid:12)(cid:12) K is constant. Proof.
Unlike the maximum principle, if the strong maximum principle is true locally, itis true globally. However, locally we have P ( G l ) ⊂ ∆ A with A >
0, so the (SMP) for ∆ A implies the (SMP) for P ( G l ).We provide an example which shows that if the core is non-empty and the equationdoes not involve all the variables, then the (MP), and hence the (SMP) can fail. Example 8.6.
Let X ⊂ R n +1 be the unit sphere S n = { ( x , ..., x n , y ) ∈ R n × R : x + · · · + x n + y = 1 } with the points y = ± H = ker (cid:0) dy (cid:12)(cid:12) T X (cid:1) be thefield of “horizontal” ( n − X tangent to the foliation by the latitudinal spheres { y = constant } , and set G l z = { H ( z ) } for z ∈ S n so that G l ⊂ G ( n − , T X ). Calculationshows that for a smooth function ϕ defined in a neighborhood of X ,(Hess X ϕ )( V, W ) = (Hess R n +1 ϕ )( V, W ) − h V, W i ν · ϕ where ν is the outward-pointing unit normal to X .Now let ϕ = (1 − y ). Then for V, W ∈ H ( z ) horizontal vector fields, the first termvanishes and the second term yields(Hess X ϕ )( V, W ) = y h V, W i Hence tr W { Hess X ϕ } = ( n − y , proving that ϕ ∈ PSH ∞ G l ( X ) and that it is G l -strictoutside y = 0. Therefore, the maximum principle fails for G l -plurisubharmonic functionson any domain Ω ⊂⊂ X which contains S n − ≡ { y = 0 } in its interior. For any suchdomain, S n − ⊂ Core(Ω)because S n − is a compact minimal G l -submanifold and therefore any G l -plurisubharmonicfunction restricted to it must be constant. (See Theorem 6.9.)22ote that tr H { Hess X u } ≥ n in local coordinates (Proposition B.3).Consequently, this subequation satisfies weak local comparison (see the discussion of theproof of Theorem 7.7). However, it does not satisfy comparison since it does not satisfythe maximum principle.We note that the maximum principle also fails for the subequation consisting of allthe p -dimensional linear subspaces of G l (given above), for any p , 1 ≤ p ≤ n −
9. Distributionally G l -Plurisubharmonic Functions. It is easy to see that for the p th partial Laplacian ∆ p on V = R n , p < n , thereare lots of distributional subharmonics (i.e., distributions u with ∆ p u a non-negative mea-sure) which are not upper semi-continuous, and hence cannot be horizontally subharmonic.However, if a closed set G l ⊂ G ( p, V ) involves all the variables, then the appropriate distri-butional definition of G l -plurisubharmonicity, although technically not equal, is equivalentto Definition 6.1. This constant coefficient result was proved in Corollary 5.4 of [HL ]. Inthis section we extend the result to the variable coefficient case.First we give the distributional definition. Definition 9.1.
A distribution u ∈ D ′ ( X ) on a riemannian manifold X is distribu-tionally G l -plurisubharmonic if ∆ A u ≥ A ( x ) of Sym ( T X ) taking values in P + ( G l ).This distributional notion can not be the “same” as G l -plurisubharmonicity, but it isequivalent in a sense we now make precise. We exclude the G l -plurisubharmonic functionswhich are ≡ −∞ on any component of X . Let L ( X ) denote the space of locally integrablefunctions on X . THEOREM 9.2.
Assume that G l ⊂ G ( p, T X ) involves all the variables and is regular.(a) Suppose u is G l -plurisubharmonic . Then u ∈ L ( X ) ⊂ D ′ ( X ) , and u isdistributionally G l -plurisubharmonic.(b) Suppose v ∈ D ′ ( X ) is distributionally G l -plurisubharmonic. Then v ∈ L ( X ) ,and there exists a unique upper semi-continuous representative u of the L ( X ) -class v which is G l -plurisubharmonic. In fact, u ( x ) = ess lim sup y → x v ( x ) is actually independent of G l . Proof.
Under the hypothesis of Theorem 9.2 we can use the next proposition along withCorollary 8.3 to reduce to proving the analogous result for A -Laplacians ∆ A where A ( x )is a smooth section of Sym ( T X ) having the additional property that A ( x ) >
0, i.e., ∆ A is uniformly elliptic. Proposition 9.3.
A distribution v ∈ D ′ ( X ) is distributionally G l -plurisubharmonic ⇐⇒ v is distributionally ∆ A -subharmonic for each smooth (local) section A of Sym ( T X ) withvalues in P + ( G l ) and A > . roof. Suppose A is a local smooth section of P + ( G l ) with A ( y ) > x ∈ X and note that since G l x involves all the variables, there exists S ∈ Int P + ( G l x )and S > G l there exists a local section S ( y ) of P + ( G l ) extending S . Since S >
0, we have that S ( y ) > U of x .Now for each ǫ >
0, ( A + ǫS )( y ) > U . That is, locally any smooth section takingvalues in P + ( G l ) can be approximated by P + ( G l )-valued sections which are positive definite.Assuming ∆ A + ǫS u ≥
0, this implies ∆ A u ≥ Completion of the Proof of Theorem 9.2.
First note that this is a local result. Notethat for each positive definite P + ( G l )-valued section A ( x ), the A -Laplacian ∆ A is of theform ∆ A u = a ( x ) · D x u + b ( x ) · D x u where a ( x ) is a positive definite n × n matrix and b ( x ) is R n -valued. Now the analogue ofTheorem 9.2, with G l -plurisubharmonicity replaced by ∆ A -subharmonicity, is true. Detailscan be found in Appendix A of [HL ]. An important point in the proof of Theorem 9.2(b) is that the upper semi-continuous representative u provided by Appendix A in [HL ]for a ∆ A -subharmonic distribution v is the same for all sections A ( x ) >
0, since it is theess-limsup regularization of the L -class v . Remark 9.4.
The ∆ A -harmonics are smooth, and the notion of ∆ A -subharmonicity isalso equivalent to the self-defining notion “sub-the-∆ A -harmonics” – again see AppendixA in [HL ].The following gives an easily verified criterion for the regularity of G l . Exercise 9.5.
Suppose G l ⊂ G ( p, T X ) is a closed subset which is a smooth fibre-wiseneighborhood retract in G ( p, T X ) . Then G l is regular. Also note that G l is a smooth fibre-wise neighborhood retract in G ( p, T X ) if and onlyif it is a smooth fibre-wise neighborhood retract in Sym ( T X ). Strictness
Recall that G l -strictness for u ∈ USC( X ) was defined in Section 7. The requirementwas that locally there exists c > u c -strict as defined by (7.8). Corollary 8.3 extendsto c -strictness as follows. Proposition 9.6.
A function u ∈ USC( X ) is c -strictly G l -plurisubharmonic ⇐⇒ u is a c -strict ∆ A -subharmonic function for each smooth (local) section A of P + ( G l ) with A > at each point. By u is c -strict for ∆ A we mean that at each point x and for each viscosity testfunction ϕ for u at x , we have (∆ A ϕ )( x ) ≥ c .A distribution v ∈ D ′ ( X ) is said to be c -strict for ∆ A (an A ≥ A v ≥ c (as an inequality of measures) . (9 . A of P + ( G l ), then v is c -strict as aG l -plurisubharmonic distribution . Proposition 9.3 easily extends to24 roposition 9.7. A distribution v ∈ D ′ ( X ) is c -strict for G l ⇐⇒ v is c -strict for ∆ A for each smooth section A of P + ( G l ) which is positive definite. Since c -strictness for the A -Laplaican, when A is positive definite, can be show to beequivalent whether interpreted with viscosity test functions or distributional test functions,Theorem 9.2 has a obvious extension to the c -strict case ( c > THEOREM 9.8.
In either part (a) or part (b) of Theorem 9.2, if the function in thehypothesis is assumed to be c -strict, one has c -strictness in the conclusion. Finally we state a result, due to Richberg [R] in the complex case, which carries overto the G l -plurisubharmonic case, assuming the following local approximation is possible. Definition 9.9.
We say that G l has the local C ∞ -approximation property if eachpoint x ∈ X has a neighborhood U such that for all u ∈ C ( U ) ∩ PSH G l ( U ) which are c -strict, and all compact K ⊂ U and ǫ >
0, there exists e u ∈ PSH ∞ G l ( U ) which is c -strict,with u ≤ e u ≤ u + ǫ on K . THEOREM 9.10.
Suppose G l has the local C ∞ strict approximation property, and let c, ǫ ∈ C ( X ) be any given continuous functions satisfying c > and > ǫ > on X . If u ∈ C ( X ) ∩ PSH G l ( X ) is c -strict, then there exists e u ∈ PSH ∞ G l ( X ) , which is (1 − ǫ ) c -strict,with u ≤ e u ≤ u + ǫ on X. The proof in Chapter I, Section 5 of [D], given in the complex case, carries over tothis much more general case. (See also [GW].)
Appendix A. Geometric Subequations
Let X be a riemannian manifold and consider a closed subset G l ⊂ G ( p, T X )of the Grassmannian of tangent p -planes. The natural candidate for a subequation F = F ( G l ) associated with G l is defined by its fibres F x = { A ∈ Sym ( T x X ) : tr W A ≥ ∀ W ∈ G l x } . ( A. W ∈ G l x the condition tr W A ≥ F x is a closed cone with vertex at the origin , and ( A. F x = { A ∈ Sym ( T x X ) : tr W A > ∀ W ∈ G l x } . ( A. P x denote the set of non-negative elements in Sym ( T x X ). Since tr W P ≥ W ∈ G ( p, T x X ) when P ∈ P x , the fibres F x defined by (A.1) satisfy the important positivity condition F x + P x ⊂ F x . ( P )Therefore the fibres F x satisfy all of the properties of a constant coefficient (euclidean)pure second-order subequation. Proposition A.1. (1) F x + Int P x = Int F x (2) F x = Int F x (3) Int F x + P x = Int F x (4) A ∈ Int F x ⇐⇒ there exists a neighborhood of A in F x of the form N ǫ ( A ) ≡ A − ǫI + Int P x for some ǫ > . Proof. (4) Note that N ǫ ( A ) is an open set containing A , and that if A − ǫI ∈ F x , thenthe positivity condition (P) implies that N ǫ ( A ) ⊂ F x .(1) By positivity F x + Int P x ⊂ F x , and it is open since it is the union over A ∈ F x ofopen sets. Hence it is contained in Int F x . Finally, Int F x ⊂ F x + Int P x by (4).(2) If A ∈ F x , then by (1) we have A + ǫI ∈ Int F x for all ǫ >
0. Hence, A =lim ǫ →∞ ( A + ǫI ) ∈ Int F x proving that F x ⊂ Int F x . Since F x is closed, we have equality.(3) The containment “ ⊂ ” is proved as in the first half of (1). The containment “ ⊃ ”follows from 0 ∈ P x .Recall the following definition from [HL ]. Definition A.2.
A (general) subset F ⊂ Sym ( T X ) is called a subequation if it satisfiesthe positivity condition: F x + P x ⊂ F x for all x ∈ X ( P )and the three topological conditions: (T ) F = Int F , (T ) F x = Int F x , (T ) Int F x = (Int F ) x . (Here Int F x means the interior relative to the fibre Sym ( T x X ).)Although F ( G l ) is not always closed (See Proposition A.6), we shall see that conditions(T ) and (T ) are always true. They will be a consequence of the following half of (T ). Lemma A.3.
The condition (T ) ′ Int F x ⊂ (Int F ) x holds for any closed subset G l ⊂ G ( p, T X ) . Consequently, if a smooth function is G l -strictat a point, then it is G l strict in a neighborhood of that point. Corollary A.4.
The set F = F ( G l ) satisfies (T ) ′ F ⊂ Int
F , (T ) F x = Int F x , (T ) Int F x = (Int F ) x . Proof.
Condition (T ) ′ implies (T ) since (Int F ) x is an open subset of F x , and hencecontained in Int F x . Property (T ) is just condition (2) in Proposition A.1. Finally, by(T ) and (T ) we have F x = Int F x = (Int F ) x ⊂ Int F which proves (T ) ′ .We can characterize the case where F ( G l ) is closed. Definition A.5.
The restricted projection π : G l → X is a local surjection if for each W ∈ G l and each neighborhood U of W , the image π ( U ∩ G l ) contains a neighborhood of π ( W ). In this case we say that G l has the local surjection property . Proposition A.6. F ( G l ) is closed ⇐⇒ π : G l → X is a local surjection. The proof is given at the end of this appendix.
Corollary A.7.
A closed subset G l ⊂ G ( p, T X ) determines a subequation F ( G l ) via (A.1)if and only if π : G l → X is a local surjection. Consequently, we adopt the following definition.
Definition A.8.
A subset F ⊂ Sym ( T X ) is a geometrically determined sube-quation if F = F ( G l ) with G l a closed subset of Sym ( T X ) having the local surjectionproperty.
Strictness
The concept of strictness given in Definition 7.8 plays an important role for uppersemi-continuous functions, not just smooth functions (see Definition (2.1) ′ ) where thenotion is unambiguous. Definition A.9. ( c -Strict). For each c > F c = F c ( G l ) to be the subset ofSym ( T X ) with fibres F cx ≡ { A ∈ Sym ( T x X ) : tr W A ≥ c ∀ W ∈ G l x } . ( A. I is a well defined smooth section of Sym ( T X ), and tr W I = p for all W ∈ G ( p, T X ). Therefore, F c = F + cp · I (fibrewise sum) . ( A. F remain true for F c ( c ≥ THEOREM A.10. If G l ⊂ G ( p, T X ) is a closed subset with the local surjection property,then for each c ≥ the set F c ( G l ) is a subequation. roofsProof of Lemma A.3. Assume we are working in a local trivialization Sym ( T ∗ V ) ∼ = V × Sym ( R n ) over an open subset V ⊂ X containing x . Then each A ∈ Sym ( T x V )determines a smooth section (also denoted A ) over V . It suffices to prove the followingtwo claims. Claim 1:
Given A ∈ Sym ( T x V ), there exists c > A ∈ Int F x ⇒ A ∈ F cy for y near x. Proof.
If not, there exist sequences { y j } in U and W j ∈ G l y j such thatlim j →∞ y j = x and lim j →∞ tr W j A = 0 . By compactness we can assume that W j → W ∈ G l x , and by continuity this gives tr W A = 0,contradicting our assumption that A ∈ Int F x (see (A.3)). Claim 2: If A is a continuous section of Sym ( T V ) and if for some c > A ( y ) ∈ F cy forall y near x , then A ( x ) ∈ Int F . Proof.
Since A ( y ) ∈ F cy , setting ǫ = cp , we have that B ( y ) ≡ A ( y ) − ǫI ∈ F y for all y near x . The set N ≡ B + Int P , defined using fibre-wise sum, is the translation of the opensubset Int P of V × Sym ( R n ) by a continuous section. Hence, N is open in Sym ( T V ).Since B ( y ) ∈ F y for all y , we have N ≡ B + Int P ⊂ F . Hence, N ⊂
Int F . Finally, A ( x ) = B ( x ) + ǫI ∈ N by positivity. Proof of Proposition A.6.
The assertion is local so we may assume that X is anopen subset of R n and π : X × G ( p, R n ) → X is projection onto the first factor, with G ( p, R n ) ⊂ Sym ( R n ).Suppose π (cid:12)(cid:12) G l is locally surjective. Let ( x j , A j ) ∈ F be a convergent sequence, x j → x, A j → A . Fix W ∈ G l x . By hypothesis for each neighborhood N δ ( W ) of W , π { ( X × N δ ( W )) ∩ G l } contains a neighborhood of x . Hence we may pick W j ∈ G l x j with W j → W .Since tr W j A j ≥ j we have tr W A ≥
0, and so A ∈ F x .For the converse, suppose π (cid:12)(cid:12) G l is not locally surjective. Then there exists ( x, W ) ∈ G land a neighborhood N ( W ) of W in G ( p, R n ) so that π { ( X × N ( W )) ∩ G l } does not containa neighborhood of x . Hence there exists a sequence of points x j → x in X , such that G l x j ∩ N ( W ) = ∅ for all j .If ǫ > V ∈ G ( p, R n ) h P V , P W ⊥ i < ǫp ⇒ V ∈ N ( W ) . ( A. h P V , − P W i ≥ − p , we have that V / ∈ N ( W ) ⇒ h P V , − P W + ǫ P W ⊥ i ≥ . ( A. G l x j ∩ N ( W ) = ∅ , this proves that A ≡ − P W + ǫ P W ⊥ ∈ F x j . However, h A, P W i = − W ∈ G l x , and so A / ∈ F x . We conclude that F is not closed.28 ppendix B. The Linear Geometric Case. In this appendix we consider the extreme geometric case where each G l x is a singlepoint W x ∈ G ( p, T x X ), or equivalently, each P ( G l x ) is the half space in Sym ( T x X ) withinward normal P W x (orthogonal projection onto W x ). Said differently, the subequation P ( G l ) is linear and given by the W -Laplacian (∆ W u ) ( x ) = h P W x , Hess x u i riem . = tr W x Hess x u. ( B. W -subharmonic functions, rather than G l -plurisubharmonicfunctions in this linear-geometric case . Example B.1. (The p th Horizontal Laplacian).
In this example, choose a single p -plane W ∈ G ( p, R n ), which might as well be the first coordinate p -plane W ≡ R p × { } ⊂ R n . Abbreviate P R p ×{ } to P . Then∆ P u = h P, D u i = tr P D u = p X j =1 ∂ u∂x j ( B. p th horizontal Laplacian . The terminology “horizontally subharmonic” and “hor-izontally p -convex” is appropriate in this case.Suppose h and H are smooth functions defined on an open subset on R n , with h taking values in GL n ( R ) and with H taking values in Hom( R n , Sym ( R n )). Definition B.2.
An equation of the form Lu = h h t P h, D u i + h H t ( P ) , Du i ( B. jet equivalent to ∆ p .The linear-geometric case is jet equivalent to ∆ p in any local coordinate system. Proposition B.3. If W is a smooth section of the Grassmann bundle G ( p, T X ) over X , then the W -Laplacian is jet equivalent to the p th horizontal Laplacian over any localcoordinate chart. Proof.
Choose a local orthonormal frame field e , ..., e n for R n with e , ..., e p a frame for W . Define h ( x ) with values in GL n ( R ) by e = h ∂∂x . Then, in the given local coordinates,∆ W u = h h t P h, D u i − h Γ t ( h t P h ) , Du i ( B. ]. Proposition B.4.
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