PLURISUBHARMONICITY and PSEUDOCONVEXITY IN CALIBRATED (and other) - PDF document

PLURISUBHARMONICITY and PSEUDOCONVEXITY IN CALIBRATED (and other) GEOMETRIES with REESE HARVEY 1

Complex Geometry: � from O or H Many Powerful techniques : also from PSH 2

Other Geometries: • Calibrated Geometries Arising, for example, in Special Holonomy Spaces SU n Sp n Sp n · Sp 1 G 2 Spin 7 • Lagrangian / Symplectic Geometry • p-convex Riemannian Geometry In ALL of these cases, PLURISUBHARMONIC FUNCTIONS • MAKE SENSE, • HAVE GOOD PROPERTIES, • ARE USEFUL FOR SOLVING PROBLEMS. 3

AXIOMS A good family of plurisubharmonic functions on a manifold X is a set of (generalized) functions PSH ( X ) which 1. is a convex cone containing the constants, 2. is closed under weak limits. 3. has an associated laplace operator ∆ for which every plurisubharmonic function u is subharmonic: ∆ u ≥ 0. Hence, every u ∈ PSH ( X ) has a unique u.s.c. [ −∞ , ∞ )-valued, L 1 loc representative. 4. is closed under composition with convex increasing functions, i.e., if ψ ′ ( t ) ≥ 0 and ψ ′′ ( t ) ≥ 0, u ∈ PSH ( X ) ⇒ ψ ◦ u ∈ PSH ( X ) , 5. satisfies u, v ∈ PSH ( X ) ⇒ max { u, v } ∈ PSH ( X ) , 4

THE STANDARD CONSTRUCTION f ∈ C ∞ ( X ) X a riemannian manifold and (Hess f )( V, W ) ≡ V Wf − ( ∇ V W ) f Given: A bundle of distinguished p -planes G D EFINITION. ↓ X f ∈ C ∞ ( X ) is G -plurisubharmonic if E XAMPLE 1. When p = 1 and G = all tangent lines, � � tr ξ { Hess f } ≡ tr { Hess f ξ } ≥ 0 for all ξ ∈ G E XAMPLE 2. When p = 2, X complex, G = C tangent lines, E XAMPLE 3. When p = dim( X ), tr T X { Hess f } = ∆ f and f is G -plurisubharmonic ⇔ f is convex. f is G -plurisubharmonic ⇔ f is plurisubharmonic f is G -plurisubharmonic ⇔ f is subharmonic. 5

D EFINITION. RELATED CONCEPTS f ∈ C ∞ ( X ) is G -plurisubharmonic if tr ξ { Hess f } ≥ 0 for all ξ ∈ G f ∈ C ∞ ( X ) is strictly G -plurisubharmonic if tr ξ { Hess f } > 0 for all ξ ∈ G f ∈ C ∞ ( X ) is G -pluriharmonic if tr ξ { Hess f } = 0 for all ξ ∈ G f ∈ C ∞ ( X ) is partially G -pluriharmonic if f is G -psh and tr ξ { Hess f } = 0 for some ξ ∈ G at every point N OTE . In general the pluriharmonic functions are scarce. However, the plurisubharmonic functions are abundant. So also are the partially pluriharmonic functions. 6

TWO FRAMEWORKS G or ( p, X ) ≡ oriented tangent p − planes ξ ∩ Λ p TX given by the unit simple vectors ξ = e 1 ∧ · · · ∧ e p . G ( p, X ) ≡ all tangent p − planes ξ ∩ Sym 2 ( TX ) given by orthogonal projection P ξ onto ξ . tr ξ A = � P ξ , A � 7

D EF. A calibration is a smooth p -form φ ∈ E p ( X ) with CALIBRATED GEOMETRY (i) dφ = 0, φ ( ξ ) ≤ 1 for all unit simple p -vectors ξ . (ii) S ET. � � � � ξ ≤ vol ξ φ D EF. An oriented p -dimensional submanifold M ⊂ X is a G ( φ ) ≡ { ξ : φ ( ξ ) = 1 } The φ -Grassmann bundle φ -submanifold if T x M ∈ G ( φ ) for all x ∈ M. � � � � M = vol M φ 8

T HEOREM . φ -submanifolds are homologically volume-minimizing . That is: If M is a φ -submanifold and M ′ is any other C 1 -submanifold with ∂M ′ = ∂M M ′ = M in H p ( M ) , and then vol( M ) ≤ vol( M ′ ) with equality if and only if M ′ is also a φ -submanifold. This extends to all rectifiable p -currents. 9

D EF. A function f ∈ C ∞ ( X ) is G ( φ ) -plurisubharmonic if T HEOREM . If f ∈ PSH ( X, φ ) , then for every φ -submanifold tr ξ { Hess f } ≥ 0 for all ξ ∈ G ( φ ) . M ⊂ X � � φ M is subharmonic N OTE . If there exist enough φ -submanifolds, the converse in the induced riemannian metric. also holds, i.e. � � f ∈ PSH ( X, φ ) ⇔ M is subharmonic ∀ M φ 10

D EFINE IMPORTANT CASE ∇ φ = 0 d φ : C ∞ ( X ) → E p − 1 ( X ) by T HEN for ξ ∈ G ( φ ) d φ f ≡ ∇ f l φ and consider dd φ : C ∞ ( X ) → E p ( X ) A ND SO ( dd φ f )( ξ ) = tr ξ Hess f ( dd φ f )( ξ ) ≥ 0 f ∈ PSH ( X, φ ) ⇔ ∀ ξ ∈ G ( φ ) . N OTE . If φ = ω = the K¨ ahler form, then d ω = d c (conformal invariance) and so dd ω = dd c . 11

In R n with ∇ φ = 0 Interesting Fact: φ = dd φ � 1 2 � x � 2 � E XAMPLE (Special Lagrangian). (like the K¨ ahler potential). in C n = R 2 n . φ ≡ Re( dz 1 ∧ · · · ∧ dz n ) Let Z ij be the form obtained from dz = dz 1 ∧ · · · ∧ dz n by replacing dz i with d ¯ z j (in the i th position). Then � � n � ∂ 2 f dd φ f = 2Re Z ij + (∆ f ) φ ∂ ¯ z i ∂ ¯ z j i,j =1 12

SYMPLECTIC GEOMETRY ( X, ω ) a symplectic manifold. ( X, ω, �· , ·� , J ) an associated Gromov manifold. ω ( v, w ) = � Jv, w � There are two important cases: Case 1. G = the J -complex 2-planes, i.e. G = G ( ω ) for the calibration ω . Note incidentally that The ω -submanifolds are exactly the J -holomorphic curves. Case 2. G ≡ LAG = the Lagrangian n-planes . Here we will have: Lagrangian plurisubharmonic functions Lagrangian convexity A Lagrangian analogue of the Monge-Amp` ere operator. 13

T HEOREM . If f is G ( ω ) -plurisubharmonic , then the restric- Case 1. G = the J -complex 2-planes, i.e. G = G ( ω ) for the calibration ω . The ω -submanifolds are exactly the J -holomorphic curves. tion of f to every pseudo-holomorphic curve is subharmonic. T HEOREM . If f is LAG -plurisubharmonic , then the restric- Case 2. G ≡ LAG = the Lagrangian n-planes . tion of f to every minimal Lagrangian submanifold is sub- harmonic. NOTE: dd c is not a good operator in the almost complex case. However, our notion of plurisubharmonicity works well. 14

RIEMANNIAN GEOMETRY P-CONVEXITY Let X be a riemannian n -manifold. Fix p , 1 ≤ p ≤ n − 1. Set G ≡ G ( p, X ) Here we will have N OTE . p-plurisubharmonic functions PSH ( X, p ) p-convexity A p -analogue of the Monge-Amp` ere operator. � f is subharmonic on all f ∈ PSH ( X, p ) ⇔ p dim ′ l minimal submanfolds E XAMPLE (p=1). This is convex geometry . Here we have convex functions standard convexity The standard (real) Monge-Amp` ere operator. 15

IN ALL THESE CASES THE AXIOMS HOLD The smooth G -plurisubharmonic functions extend to a good family of generalized functions PSH ( X ) which 1. is a convex cone containing the constants, 2. is closed under weak limits. 3. has an associated laplace operator ∆ for which every plurisubharmonic function u is subharmonic: ∆ u ≥ 0. Hence, every u ∈ PSH ( X ) has a unique u.s.c. [ −∞ , ∞ )-valued, L 1 loc representative. 4. is closed under composition with convex increasing functions, i.e., if ψ ′ ( t ) ≥ 0 and ψ ′′ ( t ) ≥ 0, u ∈ PSH ( X ) ⇒ ψ ◦ u ∈ PSH ( X ) , 5. satisfies u, v ∈ PSH ( X ) ⇒ max { u, v } ∈ PSH ( X ) , The calibrations φ that have properties 2 and 3 must satisfy: φ involves all the variables φ can be written as a positive linear combination of φ -planes at each point. ELLIPTIC CALIBRATIONS 16

IN ALL THESE CASES THE FOLLOWING THEOREMS HOLD Assume X is a riemannian manifold with a good family of plurisubharmonic functions: PSH ( X ) Moreover, assume this comes from a standard construction G → X 17

D EFINITION. If K ⊂⊂ X , the G -convex hull of K is CONVEXITY Suppose X is a non-compact. T HEOREM . The following two conditions are equivalent. � K ≡ { x ∈ X : f ( x ) ≤ sup f for all f ∈ PSH ( X ) } K If K ⊂⊂ X , then � K ⊂⊂ X . 1) 2) There exists a G -plurisubharmonic proper exhaustion function f on X . Such manifolds are called G -convex . 18

T HEOREM . The following two conditions are equivalent: � 1) K ⊂⊂ X ⇒ K ⊂⊂ X , and X carries some strictly G -plurisubharmonic function. 2) There exists a strictly G -plurisubharmonic proper exhaustion function for X . T HEOREM . The following two conditions are equivalent: Such manifolds are called strictly G -convex . � 1) K ⊂⊂ X ⇒ K ⊂⊂ X , and there exists a strictly G -plurisubharmonic function defined outside a compact subset. 2) There exists a proper exhaustion function on X which is strictly G -plurisubharmonic outside a compact subset. Such manifolds are strictly G -convex at infinity. 19

THE CORE of X D EFINITION . The core of X is the intersection For f ∈ PSH ( X ), define W ( f ) ≡ { x ∈ X : f is partially pluriharmonic at x } . � Core( X ) ≡ W ( f ) P R OPOSITION . For a calibrated manifold ( X, φ ) , every com- f ∈ P SH ( X ) T HEOREM . Suppose X is G -convex. Then: pact φ -submanifold M ⊂ X is contained in the core. Core( X ) is compact iff X is strictly G -convex at ∞ , Core( X ) = ∅ iff X is strictly G -convex. 20

D EFINITION . Suppose ρ is a defining function for ∂ Ω, i.e., BOUNDARY CONVEXITY. Let Ω ⊂⊂ X be open with smooth boundary ∂ Ω. ρ is smooth on a neighborhood of Ω with Ω = { x : ρ ( x ) < 0 } . Then ∂ Ω is called G -convex if tr ξ Hess ρ ≥ 0 for all ξ ∈ G tangent to ∂ Ω If tr ξ Hess ρ > 0, all such ξ , ∂ Ω is called strictly G -convex . This is independent of the choice of defining function ρ . P R OPOSITION . Let B denote the second fundamental form of ∂ Ω with respect to the outward pointing normal. Then ∂ Ω is G -convex if and only if � � � � ≤ 0 tr B ξ for all G -planes ξ which are tangent to ∂ Ω . 21