p-convexity, p-plurisubharmonicity and the Levi problem
aa r X i v : . [ m a t h . DG ] M a r P-CONVEXITY, P-PURISUBHARMONICTYAND THE LEVI PROBLEMF. Reese Harvey and H. Blaine Lawson, Jr. ∗ ABSTRACT
Three results in p -convex geometry are established.First is the analogue of the Levi problem in several complexvariables, namely: local p -convexity implies global p -convexity.The second asserts that the support of a minimal p -dimensionalcurrent is contained in the p -hull of the boundary union withthe “core” of the space. Lastly, the exteme rays in the convexcone of p -positive matrices are characterized. This is a basicresult with many applications. TABLE OF CONTENTS
1. Introduction.2. Plurisubharmonicity.3. Convexity, Boundary Convexity, and Local Convexity.4. Minimal Varieties and Hulls.5. Extreme rays in the Convex Cone P p ( V ). ∗ Partially supported by the N.S.F. 1 . Introduction.
On any riemannian n -manifold there are intrinsic notions of p -plurisubharmonicity and p -convexity for integers p between 1 and n . They interpolate between convexity ( p = 1)and subharmonicity ( p = n ) with p = n − p -geometry.The central algebraic idea is that of p -positivity for a quadratic form Q on a finite-dimensional inner product space V . By definition Q is p -positive if the trace of its restric-tion to every p -dimensional subspace W ⊂ V satisfies tr (cid:8) Q (cid:12)(cid:12) W (cid:9) ≥
0. This is equivalentto the condition that λ + · · · + λ p ≥ λ ≤ · · · ≤ λ n are the ordered eigenvaluesof Q . The set of such Q will be denoted P p ( V ). On any riemannian manifold X , a func-tion u ∈ C ( X ) is p -plurisubharmonic if its riemannian hessian is p -positive. An orientedhypersurface in X is p -convex if its second fundamental form is p -positive. The Riemanncurvature R of X is p - positive if for each tangent vector v , the quadratic form h R v, · · , v i is p -positive.The smooth p -plurisubhamonic functions are “pluri”-subharmonic in the followingsense. Theorem 2.12.
A function u ∈ C ( X ) is p -plurisubharmonic if and only if its restrictionto every p -dimensional minimal submanifold is subharmonic in the induced metric. The notion of p -plurisubhamonicity can be generalized to arbitrary upper semi-contin-uous [ −∞ , ∞ )-valued functions using standard viscosity test functions (cf. [CIL], [C]). For p = 1 , n this recaptures the classical notions of general convex and subharmonic functionson a riemannian manifold X . This family of upper semi-continuous p -plurisubharmonicfunctions, denoted PSH p ( X ), has many of the useful properties of subharmonic functions(see Theorem 2.6 in [HL ]). Moreover, the Restriction Theorem 2.12 has a non-trivialextension to general, upper semi-continuous p -plurisubharmonic functions (see [HL ]).The smooth p -plurisubharmonic functions can be used to introduce a notion of p -convexity as follows. Given a compact subset K ⊂ X , define the p -convex hull of K to bethe set b K of points x ∈ X such that u ( x ) ≤ sup K u for all smooth p -plurisubharmonic functions u on X . Then X is said to be p -convex if K ⊂⊂ X ⇒ b K ⊂⊂ X. The following result was proven in [HL ]. A riemannian manifold X is p -convex if and only if X admits a smooth p -plurisubharmonic proper exhaustion function. A domain Ω ⊂ X is said to be locally p -convex if each point x ∈ ∂ Ω has a neighborhood U such that Ω ∩ U is p -convex. Note that p -convex domains are locally p -convex (see (3.1)).The following converse is an analogue of Levi Problem in complex analysis, and is one ofthe three new results of this paper. 2 heorem 3.7. Let Ω ⊂⊂ R n be a domain with smooth boundary. If Ω is locally p -convex,then Ω is p -convex. There is also a notion of p -convexity for the boundary. Let II denote the secondfundamental form of the boundary ∂ Ω with respect to the interior normal. Then theboundary ∂ Ω is p -convex if II x is p -positive at each point x ∈ ∂ Ω. Theorem 3.9.
Let Ω ⊂⊂ R n be a domain with smooth boundary. If Ω is locally p -convex,then ∂ Ω is p -convex. From Theorem 3.10 one then concludes that for such domains Ω,Ω is p -convex ⇐⇒ Ω is locally p -convex ⇐⇒ ∂ Ω is p -convex.A quadratic form A on an inner product space V is said to be strictly p -positive iftr (cid:8) A (cid:12)(cid:12) W (cid:9) > p -planes W ⊂ V . This gives notions of strict p -plurisubharmonicity,strict p -convexity, etc. In Section 4 a number of results concerning strictly p -convex do-mains and strictly p -convex boundaries are discussed. A key concept here is that of the core of X , a subset which governs the existence of strictly p -plurisubharmonic functionsand proper exhaustions (see Remark 4.4.)The core contains all compact p -dimensional minimal submanifolds without boundaryin X . This result is extended to include non-compact minimal submanifolds and currents.A p -dimensional rectifiable current T ∈ R p ( X ) on X is called minimal if the first variationof the mass of T is zero with respect to deformations supported away from its boundary ∂T (see Definition 4.7). Corollary 4.10.
Suppose T ∈ R p ( X ) is a minimal current, and let u be any smooth p -plurisubharmonic function which vanishes on a neighborhood of supp( ∂T ) . Then tr −→ T (Hess u ) ≡ T ) . If T = [ M ] is the current associated to a connected p -dimension minimal submanifold, andif the p -plurisubharmonic function u and its gradient both vanish at points of ∂M , then u (cid:12)(cid:12) M ≡ , if ∂M = ∅ , u (cid:12)(cid:12) M ≡ constant . Our second result is the following.
THEOREM 4.11.
Let K ⊂ X be a compact subset and suppose T ∈ R p ( X ) is a minimalcurrent such that supp( ∂T ) ⊂ K . Then supp( T ) ⊂ ˆ K ∪ Core( X ) . This leads to the notion of a minimal surface hull of a compact set K ⊂ X , namely theunion of the supports of all minimal currents T ∈ R p ( X ) whose boundaries are supportedin K . Theorem 4.11 says that this hull is contained in b K ∪ Core( X ).Much of this discussion carries over to minimal (not necessarily rectifiable) p -currents.3ur third new result (see Section 5) describes the extreme rays in the convex cone P p ( V ), defined for each real number 1 ≤ p ≤ n by P p ( R n ) def = (cid:26) A ∈ Sym ( R n ) : λ ( A ) + · · · + λ [ p ] ( A ) + ( p − [ p ]) λ [ p ]+1 ( A ) ≥ (cid:27) (1 . p ] denotes the greatest integer ≤ p (cf. Remark 2.9). The endpoint cases canbe excluded from the discussion since P n ( V ) is a half-space (and hence has no extremerays) while it is well known that the extreme rays in P ( V ) = P ( V ) = { A ≥ } aregenerated by the orthogonal projections onto lines. These rays remain extreme in P p ( V )for 1 ≤ p < n −
1. Theorem 5.1c states that for 1 < p < n the only other extreme raysare generated by the elements of Sym ( V ) with one negative eigenvalue − ( p −
1) and allother eigenvalues 1.This technical result is more important than it may seem at first glance. Thesegenerators are exactly, up to a positive scale, the second derivatives of the
Riesz kernel K p ( X ), which is defined by: K p ( X ) = | x | − p if 1 ≤ p < | x | if p = 2 , and − | x | p − if 2 < p ≤ n. (1 . THEOREM 5.1a. ( < p < n ). Suppose F ⊂ Sym ( R n ) is a convex cone subequation.The Riesz kernel K p is F -subharmonic if and only if P p ( R n ) ⊂ F . This result has many applications. One important reason is that it holds for all real numbers p between 1 and n . In addition we note that: Many of the results from p -convex analysis hold for any real p , ≤ p ≤ n . Specifically, since P p ( R n ) ⊂ Sym ( R n ) is a convex cone, all the results of [HL ] apply.Finally we note that the basic notions of p-plurisubharmonicity and p-convexity alsomake sense with the grassmann bundle G(p,TX) replaced by a closed subset G l ⊂ G ( p, T X ).There are surprisingly many results which hold in the general context of a “ G l -geometry”.They are discussed in a companion paper [HL ].4 . Plurisubharmonicity. 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 . Usingthe inner product, this space is identified with the space of quadratic forms on V . Thenotion of p -plurisubharmonicity for a smooth function u on V is defined by requiring thatits hessian (i.e., second derivative D x u ) belong to a certain subset P p ( V ) ⊂ Sym ( V ). Tobetter understand this subset we offer several (equivalent) definitions. Definition 2.1.
Suppose A ∈ Sym ( V ). Then A ∈ P p ( V ), or A is p -positive , if thefollowing equivalent conditions hold.(1) tr W A ≥ W ∈ G ( p, V )(2) λ ( A ) + · · · + λ p ( A ) ≥ D A ≥ G ( p, V ) denotes the set of p -dimensional subspaces of V , and 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 ,(2) λ ( A ) ≤ · · · ≤ λ n ( A ) are the ordered eigenvalues of A , so Condition (2) says thatthe sum of the p smallest eigenvalues is ≥ D A : Λ p V → Λ p V is the linear action of A as a derivation on the space Λ p V of p -vectors, i.e., on simple p -vectors one has D A ( v ∧ · · · ∧ v p ) = ( Av ) ∧ v ∧ · · · ∧ v p + v ∧ ( Av ) ∧ · · · ∧ v p + v ∧ v ∧ · · · ∧ ( Av p ).The inner product on V induces an inner product on Λ p V , and we have D A ∈ Sym (Λ p V ), so the notions of non-negativity, D A ≥
0, and positive definiteness, D A > D A .The proof that condition (1), (2) and (3) are equivalent will be given below. Definition 2.2. ( p -plurisubharmonicity). A smooth function u defined on an opensubset X ⊂ R n is said to be p -plurisubharmonic if D x u ∈ P p ( R n ) for each point x ∈ X .The next result justifies the terminology. Proposition 2.3.
A function u ∈ C ∞ ( X ) is p -plurisubharmonic if and only if the restric-tion u (cid:12)(cid:12) W ∩ X is subharmonic for all affine p -planes W ⊂ R n . (Here “subharmonic” meansthat ∆ W (cid:0) u (cid:12)(cid:12) W ∩ X (cid:1) ≥ where ∆ W is the euclidean Laplacian on the affine subspace W ). 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 . 5 emark 2.4. The endpoint cases are classical.( p = ) Convex Functions.
Note that A ∈ P ⇐⇒ λ min ( A ) ≥ ⇐⇒ A ≥
0, so that u is 1-plurisubharmonic ⇐⇒ u is convex.( p = n ) Classical Subharmonic Functions.
Note that A ∈ P n ⇐⇒ tr A ≥
0, so that u is n -plurisubharmonic ⇐⇒ ∆ u ≥
0, i.e., u is classically subharmonic.Consequently, the simplest new case is when p = 2 in R where u is 2-plurisubharmonic ⇐⇒ the restriction of u to each affine plane in R is classically subharmonic. Onegeneralization of this case has an interesting characterization.( p = n − ) If p = n −
1, then ∗ : Λ V → Λ n − V is an isomorphism. This inducesan isomorphism Sym (Λ n − V ) → Sym (Λ V ) sending D A (tr A ) I − A. Therefore u ∈ C ∞ ( X ) is n − u ) I − Hess u ≥ . Note. (a) It is obvious from Condition (2) that P p ( V ) ⊂ P p +1 ( V ), or equivalently,if u is p -plurisubharmonic, then u is ( p + 1)-plurisubharmonic. In particular, each p -plurisubharmonic function is classically subharmonic, and every convex function is p -plurisubharmonic for all p .(b) The set P p ( V ) is a closed convex cone with vertex at the origin.The proof of the equivalence of Conditions (1), (2) and (3) in Definition 2.1 requiressome elementary facts. Note that each p -plane W ⊂ V determines a line L ( W ) ⊂ Λ p V ,namely the line through v ∧ · · · ∧ v p where v , ..., v p is any basis for W . If e , ..., e n is anorthonormal basis of V , we set e I = e i ∧ · · · ∧ e i p for I = ( i , ..., i p ) with i < i < · · · < i p . Lemma 2.5.
Given A ∈ Sym ( V ) , consider D A ∈ Sym (Λ p V ) . Then we have:(a) For all W ∈ G ( p, V ) , tr W A = tr L ( W ) D A . (2 . (b) If A has eigenvectors e , ..., e n with corresponding eigenvalues λ , ..., λ n , then D A has eigenvectors e I with corresponding eigenvalues λ I = λ i + · · · λ i p . (2 . Proof.
For (a), note that, if e , ..., e p is an orthonormal basis of W , then tr L ( W ) D A = h D A ( e ∧· · ·∧ e p ) , e ∧· · ·∧ e p i = P nj =1 h e ∧· · ·∧ Ae j ∧· · ·∧ e p , e ∧· · ·∧ e p i = P nj =1 h Ae j , e j i =tr W A . For (b), compute D A e I = λ I e I . Corollary 2.6.
Suppose A ∈ Sym ( V ) has ordered eigenvalues λ ( A ) ≤ · · · ≤ λ n ( A ) .Then inf W ∈ G ( p,W ) tr W A = λ ( A ) + · · · + λ p ( A ) = λ min ( D A ) , (2 . the smallest eigenvalue of D A . Proof.
Since D A has eigenvalues λ I by part (b), the smallest is λ ( A ) + · · · + λ p ( A ) =tr L ( W ) D A where W = span { e , ..., e p } . Now the smallest eigenvalue of D A equals the6nfimum of tr L D A over all lines in Λ p V , so in this case it is also the infimum over therestricted set of lines of the form L ( W ) with W ∈ G ( p, V ). By part (a) in Lemma 2.5, thisproves (2.3).The equivalence of Conditions (1), (2) and (3) in Definition 2.1 is immediate fromCorollary 2.6. Definition 2.7. ( p -Harmonic). A smooth function u defined on an open subset X ⊂ R n is p -harmonic if D x u ∈ ∂ P p for all x ∈ X , or equivalently if λ min ( D D x u ) = λ ( D x u ) + · · · + λ p ( D x u ) = 0 for all x ∈ X . Example 2.8. (Radial Harmonics). ( p = ) The function | x | is 1-harmonic on R n − { } .( p = ) The function log | x | is 2-harmonic on R n − { } .( ≤ p ≤ n ) The function − | x | p − is p -harmonic on R n − { } . Proof.
Given a non-zero vector x ∈ R n , let P x ≡ | x | x ◦ x denote orthogonal projectiononto the line through x . One calculates that: D | x | = | x | ( I − P x ) , (2 . D log | x | = | x | ( I − P x ) , (2 . D (cid:16) − | x | p − (cid:17) = ( p − | x | p ( I − p P x ) . (2 . u ( x ) defined in Example 2.8 has second derivative D u , which is a positive scalar multiple of H ≡ I − pP x , and that H has one negativeeigenvalue − ( p −
1) and the other eigenvalues are 1. By Lemma 2.5(b) this implies thatthe eigenvalues of D H are 0 and p, and in particular, λ min ( D H ) = 0. Remark 2.9. (Non-Integer p ). The subset (subequation) P p ( V ) can be defined forany real number p between 1 and n in such a way that many of the results in this papercontinue to hold for non-integer values of p . Let ¯ p = [ p ] denote the greatest integer in p .Then we define A ∈ Sym ( V ) to be p -positive , or A ∈ P p ( V ), if λ ( A ) + · · · + λ ¯ p ( A ) + ( p − ¯ p ) λ ¯ p +1 ≥ , (2 . λ ( A ) ≤ · · · ≤ λ n ( A ) denote the ordered eigenvalues of A . To see that P p ( V ) is a convex cone, one shows that it is the polar of the set of P e + · · · + P e ¯ p +( p − ¯ p ) P e ¯ p +1 where e , ..., e n are orthonormal.The motivation for this definition of P p is provided by the next remark and Theorem5.1. These are the only two other places in this paper where non-integer values of p arediscussed. In the other places (such as Definition 3.1) the gaps are left to the reader. Remark 2.10. (The Riesz Kernel).
The family of functions defined in Example 2.8naturally extends by (1.2) to all real numbers p between 1 and n , and we have the following.7 emma 2.11. For each real number p with ≤ p ≤ n , K p ( x ) is p harmonic on R n − { } and p plurisubharmonic on R n . Proof.
Up to a positive scalar multiple D x K p equals H = I − pP x . As noted above D H ≥ λ min ( D H ) = 0. Riemannian Manifolds.
Suppose X is an n -dimensional riemannian manifold. Then the euclidean notionsabove carry over with V = T x X and the ordinary hessian of a smooth function replacedby the riemannian hessian . For u ∈ C ( X ) this is a well defined section of the bundleSym ( 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. Acting as a derivation, it determines a welldefined section D Hess u of Sym (Λ p T X ) for each p , 1 ≤ p ≤ n . Definition 2.2 ′ ( p -plurisubharmonicity). A smooth function u on X is said to be p -plurisubharmonic if Hess x u is p -positive at each point x ∈ X (see Definition 2.1).The appropriate geometric objects for restriction are the p -dimensional minimal (sta-tionary) submanifolds of X . In the euclidean case this enlarges the family of affine p -planesused in Proposition 2.3 when 1 < p < n . THEOREM 2.12.
A function u ∈ C ( X ) is p -plurisubharmonic if and only if therestriction of u to every p -dimensional minimal submanifold is subharmonic. Proof.
Suppose M ⊂ X is any p -dimensional submanifold, and let H M denote its meancurvature vector field. Then (see Proposition 2.10 in [HL ])∆ M (cid:0) u (cid:12)(cid:12) M (cid:1) = tr T M
Hess u − H M u. (2 . M is minimal, then∆ M (cid:0) u (cid:12)(cid:12) M (cid:1) = tr T M
Hess u. (2 . x ∈ X and every p -plane W ⊂ T x X , there exists a minimal submanifold M with T x M = W . This is enoughto conclude Theorem 2.12 from (2.10). 8 . Convexity, Boundary Convexity, and Local Convexity Riemannian Manifolds
Let PSH ∞ p ( X ) denote the smooth p -plurisubharmonic functions on a riemannian man-ifold X . Definition 3.1.
Given a compact subset K ⊂ X , the p -convex hull of K is the set b K ≡ { x ∈ X : u ( x ) ≤ sup K u for all u ∈ PSH ∞ p ( X ) } Proposition 3.2. If M ⊂ X is a compact connected p -dimensional minimal submanifoldwith boundary ∂M = ∅ , then M ⊂ d ∂M . Proof.
Apply Theorem 2.12 and the maximum principle for subharmonic functions on M . Definition 3.3.
We say that X is p -convex if for all compact sets K ⊂ X , the hull b K isalso compact. THEOREM 3.4.
Suppose X is a riemannian manifold. Then:(1) X is p -convex ⇐⇒ (2) X admits a smooth p -plurisubharmonic proper exhaustion function. Proof.
See Theorem 4.4 in [HL ] for the proof. It is exactly the same proof as the onegiven for Theorem 4.3 in [HL ].Condition (2) can be weakened to a local condition at ∞ in the one-point compacti-fication X = X ∪ {∞} . This follows from the next lemma. Lemma 3.5.
Suppose that X − K admits a smooth p -plurisubharmonic function v with lim x →∞ v ( x ) = ∞ where K is compact. Then X admits a smooth p -plurisubharmonicproper exhaustion function which agrees with v near ∞ . Proof.
This is a special case of Lemma 4.6 in [HL ]. Euclidean Space.
We now show that the p -convexity of a compact domain with smooth boundary ineuclidean space is a local condition on the domain near the boundary. This result is tosome degree analogous to the Levi Problem in complex analysis, and is one of the threenew results of this paper. Definition 3.6.
A domain Ω ⊂ R n is locally p -convex if each point x ∈ ∂ Ω has aneighborhood U in R n such that Ω ∩ U is p -convex.9ach ball in R n is p -convex, and the intersection of two p -convex domains is again p -convex. Therefore If Ω is p − convex , then Ω is locally p − convex . (3 . THEOREM 3.7.
Suppose that Ω is a compact domain with smooth boundary. If Ω islocally p -convex, then Ω is p -convex. Intermediate between local and global convexity is the notion of boundary convexity.Suppose now that ∂ Ω is smooth.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 ∂ Ω ( 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 . Definition 3.8. The boundary ∂ Ω is p -convex at a point x if tr W { II ∂ Ω } ≥ p -planes W ⊂ T x ( ∂ Ω) at x .Theorem 3.7 is the compilation of the following two results. THEOREM 3.9.
If the domain Ω is locally p -convex, then its boundary ∂ Ω is p -convex. THEOREM 3.10.
If the boundary ∂ Ω is p -convex, then the domain Ω is p -convex. Before proving these two theorems we make some remarks on boundary convexity.
Remark 3.11. (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 is non-zero on ∂ Ω ∩ B , then ρ is called a local defining function for ∂ Ω. It has theproperty that D x ρ = |∇ ρ ( x ) | II x (3 . ∂ Ω ∩ B . To see this, suppose that e is a vector field 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 ) =( D ρ )( e, e ). As a consequence we have that ∂ Ω is p -convex at a point x if and only iftr W D x ρ ≥ p − planes W tangent to ∂ Ω at x (3 . ρ is a local defining function for ∂ Ω. Moreover, (3.3) is independent of the choiceof the local defining function.
Remark 3.12. (Principal curvatures).
Let κ ≤ · · · ≤ κ n − denote the orderedeigenvalues of II x . Then we have that ∂ Ω is p − convex at x ⇐⇒ κ + · · · + κ p ≥ . (3 . Proof.
Apply Corollary 2.6 to A ≡ II with V ≡ T x ∂ Ω.10e now give the proof of Theorem 3.9, that local p -convexity implies boundary p -convexity. Lemma 3.13. If ∂ Ω is not p -convex at a point x ∈ ∂ Ω , then there exists an embeddedminimal p -dimensional submanifold M through the point x with M − { x } ⊂ Ω in a neighborhood of x. (3 . Proof of Theorem 3.9 . Assume that ∂ Ω is not p -convex at a point x ∈ ∂ Ω. Let B denote the ǫ -ball about x . It suffices to show that Ω ∩ B is not p -convex. This is done byconstructing a “tin can” inside B using Lemma 3.13. We can assume that M is a compactmanifold with boundary and M ⊂ B .Let M t ≡ M + tν denote the translate of M by tν where ν is the outward-pointing unitnormal to ∂ Ω at x . Choose r > M t ⊂ Ω for − r ≤ t <
0. Let K denote the “empty tin can” consisting of the “bottom” M − r and the “label” S − r ≤ t ≤ ∂M t .Then K is a compact subset of Ω ∩ B . Let ˆ K be its p -convex hull in Ω ∩ B .Since ∂M t ⊂ K , Proposition 3.2 implies that each M t ⊂ ˆ K for − r ≤ t <
0. Since ˆ K isclosed in Ω ∩ B , this proves that x must be in the R n -closure of ˆ K , i.e., ˆ K is not compact.Hence, Ω ∩ B is not p -convex. Proof of Lemma 3.13.
Suppose ∂ Ω is not p -convex at x . Then there is a tangent p -plane W to ∂ Ω at x with tr W { II ∂ Ω } < . (3 . W is the plane spanned by eigenvectors of II with the smallesteigenvalues. We can then choose euclidean coordinates ( t , ..., t n ) with respect to an or-thonormal basis e , ..., e n so that:(i) x corresponds to the origin 0,(ii) n = e n is the outward pointing normal to Ω at x .(iii) e , ..., e n − are the eigenvectors of II at x with eigenvalues κ ≤ κ ≤ · · · ≤ κ n − (iv) W = span { e , ..., e p } In a neighborhood of 0 our domain can be written asΩ = { t n < f ( t , ..., t n − ) } . In particular, ρ ( t ) ≡ t n − f ( t , ..., t n − ) is a local defining function for ∂ Ω near 0 ∈ ∂ Ω.By Remark 3.11, since ( ∇ ρ )(0) = e n is a unit vector, D ρ = − D f = II . (3 . f has Taylor expansion f ( t ) = − ( κ t + · · · + κ n − t n − ) + O ( | t | ) . (3 . c ≡ − p ( κ + · · · + κ p ) we obtain a diagonal matrix diag( κ + c, ..., κ p + c ) withtrace zero. The hypothesis (3.6) is equivalent to c > P ≡ span { e , ..., e p , e n } = W ⊕ R e n , and consider graphs { t n = g ( t , ..., t p ) } which are minimal hypersurfaces in P (andtherefore in R n ). We apply the following basic lemma, whose proof is left as an exercise. Lemma 3.14.
Given A ∈ Sym ( R p ) with tr A = 0 , there exists a real analytic function g defined near the origin with g (0) = 0 , ( ∇ g )(0) = 0 and D g = A such that g satisfies theminimal surface equation. We can apply this lemma with A = − diag( κ + c, ..., κ p + c ) obtaining a minimalsurface M = { ( t, g ( t )) ∈ P = R p +1 : | t | < η } ⊂ R n . The hypothesis c > g ( t ) < f ( t , ..., t p , , ...,
0) if 0 < | t | < η small. This implies that M − { } ⊂ Ω, completingthe proof of Lemma 3.13 and Theorem 3.9 as well.Now we commence with the proof of Theorem 3.10. Let δ ( x ) denote the distancefrom a point x ∈ Ω to the boundary ∂ Ω. By the ǫ -collar of ∂ Ω we shall mean the set { x ∈ Ω : 0 < δ ( x ) < ǫ } . Theorem 3.10 is immediate from the next result. Proposition 3.15. (1) If ∂ Ω is p -convex on a neighborhood of x ∈ ∂ Ω , then − log δ ( x ) is p -plurisubharmonicon the intersection of a neighborhood of x in R n with an ǫ -collar of ∂ Ω .(2) If − log δ ( x ) is p -plurisubharmonic on an ǫ -collar of ∂ Ω , then Ω is p -convex. Summary 3.16.
From this proposition and Theorems 3.9 and 3.10 we conclude thatΩ is locally p − convex ⇐⇒ ∂ Ω is p − convex ⇐⇒ − log δ ( x ) is p − plurisubharmonic ⇐⇒ Ω is p − convex (3 . Proof of (1).
Let II denote the second fundamental form of the hypersurfaces { δ = ǫ } for ǫ ≥
0, and let n = ∇ δ denote the inward-pointing normal. An arbitrary p -plane V ata point can be put in a canonical form with basis(cos θ ) n + (sin θ ) e , e , ..., e p where n, e , ..., e p are orthonormal. Set W ≡ span { e , ..., e p } , the tangential part of V . Lemma 3.17. tr V Hess( − log δ ) = 1 δ sin θ tr W ( II ) + 1 δ cos θ Proof.
See Remark after Proposition 5.13 in [HL ]. Note.
This formula holds on any riemannian manifold.If II has eigenvalues κ , ..., κ n − at a point x ∈ ∂ Ω, then let κ ( δ ) , ..., κ n − ( δ ) denotethe eigenvalues of II at the point a distance δ from x along the normal line. A proof ofthe following can be found in [GT, § emma 3.18. For small δ ≥ one has κ j ( δ ) = κ j − δκ j , j = 1 , ..., n − . Corollary 3.19.
Each κ j ( δ ) is strictly increasing if κ j = 0 and ≡ if κ j = 0 . We now combine Lemma 3.17 with Corollary 3.19 to conclude that − log δ is p -plurisubharmonic. Remark 3.20.
Note that each ∂ Ω ǫ , where Ω ǫ ≡ { δ > ǫ } , is strictly p -convex and − log δ is strictly p -plurisubharmonic if and only if ∂ Ω has no p -flat points, i.e., points where thenullity of II ∂ Ω is ≥ p . Proof of (2).
By Theorem 3.4 it suffices to prove the existence of a continuous exhaustionfunction u : Ω → R + which is smooth and p -plurisubharmonic outside a compact set inΩ. Such a function is given by setting u ( x ) = max {− log δ ( x ) , − log( ǫ/ } . Remark 3.21.
It would be interesting to determine if Theorem 3.9 remains true for allreal numbers p between 1 and n . Most of the other results of this section do extend to allsuch p by [HL ].
4. Minimal Varieties and Hulls.
There are several notions of the p -convex hull of a set, all of which are intimatelyrelated to minimal currents. We begin by recalling the following. Strict Convexity.
Let X be a riemannian manifold which is connected and non-compact. Definition 4.1.
We say that a function u ∈ PSH ∞ p ( X ) is strictly p -plurisubharmonicat a point x ∈ X if Hess x u ∈ Int P p ( T x X ), i.e., if one of the following equivalent conditionsholds:(1) tr W Hess x u > W ∈ G ( p, T x X ),(2) λ (Hess x u ) + · · · + λ p (Hess x u ) > D Hess x u > λ ( A ) ≤ λ ( A ) ≤ · · · denote the ordered eigenvalues of A . Definition 4.2.
The manifold X is called strictly p -convex if it admits a properexhaustion function u : X → R which is strictly p -plurisubharmonic at every point,and it is called strictly p -convex at infinity if it admits a proper exhaustion function u : X → R which is strictly p -plurisubharmonic outside a compact subset. Definition 4.3.
The p -core of X is defined to be the subsetCore p ( X ) ≡ { x ∈ X : u is not strict at x for all u ∈ PSH ∞ p ( X ) } emark 4.4. This concept is useful in conjunction with Definition 4.2.(1) X admits a smooth strictly p -plurisubharmonic function ⇐⇒ Core( X ) = ∅ .(2) X is strictly p -convex, i.e., X admits a smooth strictly p -plurisubharmonic properexhaustion function ⇐⇒ Core( X ) = ∅ and X is p -convex.(3) X is strictly p -convex at infinity ⇐⇒ Core( X ) is compact and X is p -convex.Part (1) is a special case of Theorem 4.2 in [HL ]; Part (2) is a special case of 4.8 in[HL ]; and Part (3) is a special case of Theorem 4.11 in [HL ].We note that when X admits a strictly p -plurisubharmonic proper exhaustion func-tion, standard Morse Theory implies that X has the homotopy-type of a complex of di-mension ≤ p − Proposition 4.5.
Every compact p -dimensional minimal submanifold M without bound-ary in X is contained in Core p ( X ) . If instead the boundary ∂M = ∅ and M is connected,then M ⊂ d ∂M . Proof.
For the first assertion, apply Theorem 2.12 and the maximum principle to concludethe restriction of any smooth p -plurisubharmonic function to M is constant. The secondassertion is Proposition 3.2.This provides an analogue of the support Lemma 3.2 in [HL ]. Corollary 4.6.
Suppose M ⊂ X is a compact p -dimensional minimal submanifold withpossible boundary. Then M ⊂ d ∂M ∪ Core( X ) . Minimal Varieties and their Associated Hulls
Now we introduce the minimal current hull of a compact set K in a riemannianmanifold X , and relate it to the p -convex hull ˆ K . This second hull will be defined using thegroup R p ( X ) of p -dimensional rectifiable currents with compact support in X (cf. [F], [Si],[M], etc.). These creatures enjoy many nice properties. They can be usefully consideredas compact oriented p -dimensional manifolds with singularities and integer multiplicities,and readers unfamiliar with the general theory can think of them simply as submanifolds.Of importance here is the following general structure theorem. Associated to each T ∈ R p ( X ) is a Radon measure k T k on X and a k T k -measurable field of unit p -vectors −→ T such that for any smooth p -form ω on X , T ( ω ) = Z X ω ( −→ T ) d k T k . (4 . p -currents are the topological dual space to the space of smooth p -forms.) In particular, every T ∈ R p ( X ) has a finite mass M ( T ) = Z X d k T k . Example.
When T corresponds to integration over a compact oriented submanifold withboundary, of finite volume M ⊂ X , one has k T k = H p (cid:12)(cid:12) M ( H p = Hausdorff measure), −→ T x corresponds to the oriented tangent plane T x M , and M ( T ) = H p ( M ) = the riemannianvolume of M . Definition 4.7.
A current T ∈ R p ( X ) is called minimal or stationary if for all smoothvector fields V on X which vanish on a neighborhood of the support of ∂T , one has ddt M (( ϕ t ) ∗ T )) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = 0 , (4 . ϕ t denotes the flow generated by V on a neighborhood of the support of T .Each smooth vector field on X defines a smooth bundle map A V : T X → T X givenon a tangent vector W by A V ( W ) def = ∇ W V. (4 . D A V : Λ p T X → Λ p T X as in Section 2. Proof of thefollowing can be found in [LS] or [L].
THEOREM 4.8. (The First Variational Formula).
Fix T ∈ R p ( X ) and let V , ϕ t be as above. Then ddt M (( ϕ t ) ∗ T )) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = Z X h D A V −→ T , −→ T i d k T k = Z X tr −→ T (cid:0) A V (cid:1) d k T k . (4 . V = ∇ u for a smooth function u on X . Then A ∇ u = Hess u, (4 . T X . To see this note that hA ∇ u ( W ) , U i = h∇ W ( ∇ u ) , U i = W h∇ u, U i − h∇ u, ∇ W U i = ( W U − ∇ W U ) u = (Hess u )( W, U ). Hence, we have the follow-ing.
THEOREM 4.9. If V = ∇ u , then ddt M (( ϕ t ) ∗ T )) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = Z X tr −→ T (Hess u ) d k T k . (4 . Corollary 4.10.
Suppose T ∈ R p ( X ) is a minimal current, and let u be any smooth p -plurisubharmonic function which vanishes on a neighborhood of supp( ∂T ) . Then tr −→ T (Hess u ) ≡ T ) . f T = [ M ] where M is a compact connected minimal submanifold of dimension p , and if u is a smooth p -plurisubharmonic function such that ∇ u (cid:12)(cid:12) ∂M = 0 , then u (cid:12)(cid:12) M = constant . Proof.
The first statement follows directly from (4.2), (4.6) and the fact that tr W Hess u ≥ p planes W .If T = [ M ] for a minimal submanifold M , then tr T x M (Hess x u ) = ∆ M ( u (cid:12)(cid:12) M ) where∆ M is the Laplace-Beltrami operator of M in the induced metric (see Proposition 2.10 in[HL ]). By the First Variational Formula in the smooth case (e.g. Theorem 1.1 in [L]) weconclude that u (cid:12)(cid:12) M is harmonic on M with constant boundary values (when ∂M = ∅ ), andthe conclusion follows from the maximum principle. THEOREM 4.11.
Let K ⊂ X be a compact subset and suppose T ∈ R p ( X ) is a minimalcurrent such that supp( ∂T ) ⊂ K . Then supp( T ) ⊂ ˆ K ∪ Core( X ) . Proof.
Suppose x / ∈ ˆ K . Then by the p -plurisubharmonic analogue of Lemma 4.2 in [HL ]there exists a smooth non-negative p -plurisubharmonic function u , which is zero on aneighborhood of K and satisfies u ( x ) >
0, and furthermore, if x / ∈ Core( X ), then u can bechosen to be strict at x . Therefore, tr −→ T (Hess u ) > U of x . Sincetr −→ T (Hess u ) ≥ k T k ( U ) = 0.Hence, x / ∈ supp( T ).This result can be rephrased in terms of a second hull defined as follows. Definition 4.12.
Given a compact subset K ⊂ X , we define the minimal p -currenthull to be the set ˆ K min = [ supp( T )where the union is taken over all minimal T ∈ R p ( X ) with supp( ∂T ) ⊂ K .Note that ˆ K min contains all compact minimal oriented p -dimensional submanifoldswith boundary in K . THEOREM 4.11 ′ . ˆ K min ⊂ ˆ K ∪ Core( X ) . By Remark 4.4(1), X supports a global strictly p -plurisubharmonic function if andonly if Core( X ) = ∅ . Therefore, ˆ K min ⊂ ˆ K in this case. For example, | x | is such a globalfunction on R n . Question 4.13.
Suppose Γ ⊂ R n is a compact ( p − p -current in R n and that current is an oriented submanifold M . How close does ˆΓ come to approximating M ?16 eneral (Not Necessarily Rectifiable) Minimal Currents Much of what is said above carries over to general compactly supported currents offinite mass. These are exactly the currents which can be represented as in (4.1) with theprovision that the k T k -measurable field −→ T of unit p -vectors is no longer required to besimple k T k -a.e.. Definition 4.7 makes sense for such currents, and the first variationalformula ddt M (( ϕ t ) ∗ T )) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = Z X h D A V −→ T , −→ T i d k T k holds. If V = ∇ u where u ∈ PSH ∞ p ( X ), then by Definition 2.1 (3) we know that D A V ≥ u is strict, we have D A V >
0. The arguments for Corollary4.10 and Theorem 4.11 give the following.
Proposition 4.14.
Let T be a minimal p -dimensional current of finite mass and compactsupport on X , and let u be any smooth p -plurisubharmonic function which vanishes on aneighborhood of supp( ∂T ) . Then h D Hess u −→ T , −→ T i = 0 k T k − a.e. Furthermore, supp( T ) ⊂ c ∂T ∪ Core( X ) . Thus the minimal current hull b K min can be expanded to contain the supports of all minimalcurrents with boundary supported in K , and Theorem 4.11 ′ remains true. Examples.
Minimal non-rectifiable currents abound in geometry. Any positive ( p, p )-current on a K¨ahler manifold X is minimal. This observation extends to positive φ -currents on any calibrated manifold ( X, φ ) (see [HL ]). Any foliation current whose leavesare minimal p -submanifolds is a minimal current.There are two basic cases of smooth minimal currents which are interesting. Let T bea smooth d -closed current of dimension n − T is simply a closed 1-formand can be written locally as T = df for a smooth function f . In a neighborhood of anypoint where df = 0, the minimality condition is equivalent to the 1-Laplace Equation: d (cid:18) ∗ df k df k (cid:19) = 0 . which says that ∗ df k df k calibrates the level hypersurfaces of f . In particular, the level setsof f are minimal varieties.Let T be a smooth d -closed current of dimension 1. Then T can be expressed on acompactly supported 1-form α as T ( α ) = R X α ( V ) d vol X where V is a smooth vector field.Minimality is the condition that ∇ V (cid:18) V k V k (cid:19) = 0 , which means exactly that the (reparameterized) flow lines of V are geodesics in X , andthe d -closed condition is equivalent to div( V ) = 0 . . The Extreme Rays in the Convex Cone P p ( V ) . Recall the classical fact that the extreme rays in P ( V ) ≡ { A : A ≥ } are exactlythose generated by the orthogonal projections P e onto the lines spanned by unit vectors e ∈ R n . The purpose of this section is to describe the extreme rays in P p ( V ) for other p .Note that P n ( V ) can be excluded from the discussion since it is a closed half-space, andhence has no extreme rays. First we state our result in ways that are more suitable for themany applications (see [HL ] and [HL ] ). THEOREM 5.1a. ( < p < n ). The convex cone P p ( R n ) ⊂ Sym ( R n ) is the smallestconvex cone subequation F with the property that the Riesz kernel K p is F -subharmonic. The second version requires a definition.
Definition 5.2.
The
Riesz characteristic p F of a subequation F ⊂ Sym ( R n ) is definedto be p F ≡ { p : I − pP e ∈ F for all | e | = 1 } . THEOREM 5.1b. ( < p < n ). Suppose that F ⊂ Sym ( R n ) is a convex conesubequation. Then P p ⊂ F ⇐⇒ p ≤ p F . Finally we state the result in terms of extreme rays.
THEOREM 5.1c. ( < p < n ). The extreme rays in P p ( V ) are of two types. They aregenerated by either (1) I − p ( e ◦ e ) = P e ⊥ − ( p − P e or (2) P e where e is a unit vector in V . If n − ≤ p < n , only case (1) occurs. Proof of Theorem 5.1c.
Under the action of O n on Sym ( R n ), the set D ≡ R n ofdiagonal matrices form an n -dimensional cross-section. For any O n -invariant set F ⊂ Sym ( R n ), the intersection F ≡ F ∩ D has orbit O ( F ) = F. For a convex cone F ⊂ Sym ( R n ), let E xt ( F ) denote the union of the extreme rays in F Lemma 5.2. If F ⊂ Sym ( R n ) is an O n -invariant convex cone and F ≡ F ∩ D , then E xt ( F ) ⊆ O ( E xt ( F )) . Proof.
Suppose
A / ∈ O ( E xt ( F )). Then A = gDg t with g ∈ O n implies D / ∈ E xt ( F ). Thus D = αD + βD with α > β > D , D ∈ F , but D and D determine different rays.Therefore, A = αgD g t + βgD g t = αA + βA , A , A ∈ O ( F ) = F , but A and A determine different rays, proving that A / ∈ E xt ( F ).In particular, E xt ( P p ) ⊂ O ( E xt ( P p )), so that it remains to compute the extreme raysin P p ≡ P p ∩ D . First note that by definition (see Remark 2.9) we have P p = P p = { A = diag( λ , ..., λ n ) : λ ↑ + · · · + λ ↑ ¯ p + ( p − ¯ p ) λ ↑ ¯ p +1 ≥ } (5 . λ ↑ ≤ λ ↑ ≤ · · · λ ↑ n denotes the rearrangement of the λ i ’s into ascending order. Lemma 5.4. If A ∈ P p is extreme, then A has at most one strictly negative eigenvalue. Proof.
Suppose λ ↑ = λ ↑ ( A ) <
0. To simplify notation we assume the λ i ’s are in ascendingorder and drop the arrows. Set α = λ + λ < λ ≡ ( λ , ..., λ n ). Then λ = sv + (1 − s ) w where s = λ α >
0, 1 − s = λ α > v = ( α, , λ , ..., λ n ), w = (0 , α, λ , ..., λ n ),and v, w ∈ P p . Hence, A is not extreme.We are now reduced to two cases. One Negative Eigenvalue:
By rescaling and permuting we may assume λ = − ≤ λ ≤ · · · ≤ λ n where A = diag( λ , λ , ..., λ n ). Set B = diag(0 , λ ..., λ n ). Then λ + · · · + λ ¯ p + ( p − ¯ p ) λ ¯ p +1 ≥ . (5 . B is extreme in the set of matrices satisfying (5.2), then λ = · · · = λ n = µ and equalityholds in (5.2). Therefore, ( ¯ p − µ + ( p − ¯ p ) µ = 1, that is, µ = 1 / ( p − A ∈ P p is extreme and has one strictly negative eigenvalue, then afterrescaling A and permuting coordinates, A = diag( − ( p − , , ..., All Eigenvalues Positive:
Consider the hyperplane λ + · · · + λ ¯ p + ( p − ¯ p ) λ ¯ p +1 = 1intersected with the positive quadrant in R ¯ p +1 (or R ¯ p if p = ¯ p ). The cone on this set isthe positive quadrant. Therefore, the only extreme rays of P p that could possibly appearfrom this set are the axis rays.This proves that the only possible extreme rays in P p ( V ) are generated by P e and I − pP e with | e | = 1. By the orthogonal invariance of P p ( V ) the ray generated by I − pP e , forone unit vector e , is extreme if and only if it is extreme for all unit vectors. Consequently,If I − pP e is not extreme for one e , then the only possible extreme rays are generated bythe rank-one projections P e . Now p < n implies P p ( V ) ∩ { A : tr A = 1 } is compact, sothat the extreme rays must generate P p ( V ). This forces P p ( V ) ⊂ P ( V ) which contradicts1 < p . Summarizing, we have that each I − pP e generates an extreme ray in P p ( V ).It remains to show that the axis rays are extreme in P p ( V ) if and only if 1 < p < n − D x K p is, up to a positive scalar, equal to I − pP x . The equivalence toversion b) is straightforward. References [C] M. G. Crandall,
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