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

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

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Complex Geometry: Many Powerful techniques :

• from O or H

also from PSH 2

Other Geometries:

• Calibrated Geometries

Arising, for example, in Special Holonomy Spaces SUn Spn Spn · Sp1 G2 Spin7

• Lagrangian / Symplectic Geometry
• p-convex Riemannian Geometry

In ALL of these cases, PLURISUBHARMONIC FUNCTIONS

• MAKE SENSE,
• HAVE GOOD PROPERTIES,
• ARE USEFUL FOR SOLVING PROBLEMS.

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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, L1

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 X a riemannian manifold and f ∈ C∞(X) (Hessf)(V, W) ≡ V Wf − (∇V W)f Given: A bundle of distinguished p-planes G ↓ X

f ∈ C∞(X) is G-plurisubharmonic if trξ{Hessf} ≡ tr{Hessf

• ξ} ≥ 0

for all ξ ∈ G

• 1. When p = 1 and G = all tangent lines,

f is G-plurisubharmonic ⇔ f is convex.

• 2. When p = 2, X complex, G = C tangent lines,

f is G-plurisubharmonic ⇔ f is plurisubharmonic

• 3. When p = dim(X), trT X{Hessf} = ∆f and

f is G-plurisubharmonic ⇔ f is subharmonic. 5

RELATED CONCEPTS

f ∈ C∞(X) is G-plurisubharmonic if trξ{Hessf} ≥ 0 for all ξ ∈ G f ∈ C∞(X) is strictly G-plurisubharmonic if trξ{Hessf} > 0 for all ξ ∈ G f ∈ C∞(X) is G-pluriharmonic if trξ{Hessf} = 0 for all ξ ∈ G f ∈ C∞(X) is partially G-pluriharmonic if f is G-psh and trξ{Hessf} = 0 for some ξ ∈ G at every point

However, the plurisubharmonic functions are abundant. So also are the partially pluriharmonic functions. 6

TWO FRAMEWORKS Gor(p, X) ≡ oriented tangent p−planes ξ ∩ ΛpTX given by the unit simple vectors ξ = e1 ∧ · · · ∧ ep. G(p, X) ≡ all tangent p−planes ξ ∩ Sym2(TX) given by orthogonal projection Pξ onto ξ. trξA = Pξ, A 7

CALIBRATED GEOMETRY

• DEF. A calibration is a smooth p-form φ ∈ Ep(X) with

(i) dφ = 0, (ii) φ(ξ) ≤ 1 for all unit simple p-vectors ξ.

• φ
• ξ ≤ volξ
• SET.

G(φ) ≡ {ξ : φ(ξ) = 1} The φ-Grassmann bundle

• DEF. An oriented p-dimensional submanifold M ⊂ X is a

φ-submanifold if TxM ∈ G(φ) for all x ∈ M.

• φ
• M = volM
• 8

φ-submanifolds are homologically volume-minimizing. That is: If M is a φ-submanifold and M ′ is any other C1-submanifold with ∂M ′ = ∂M and M ′ = M in Hp(M), 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

• DEF. A function f ∈ C∞(X) is G(φ)-plurisubharmonic if

trξ{Hessf} ≥ 0 for all ξ ∈ G(φ).

M ⊂ X φ

• M is subharmonic

in the induced riemannian metric.

also holds, i.e. f ∈ PSH(X, φ) ⇔ φ

• M is subharmonic ∀M

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IMPORTANT CASE ∇φ = 0

dφ : C∞(X) → Ep−1(X) by dφf ≡ ∇f l φ and consider ddφ : C∞(X) → Ep(X)

(ddφf)(ξ) = trξHessf

f ∈ PSH(X, φ) ⇔ (ddφf)(ξ) ≥ 0 ∀ξ ∈ G(φ).

ahler form, then dω = dc (conformal invariance) and so ddω = ddc. 11

Interesting Fact: In Rn with ∇φ = 0 φ = ddφ 1

2x2

(like the K¨ ahler potential).

φ ≡ Re(dz1 ∧ · · · ∧ dzn) in Cn = R2n. Let Zij be the form obtained from dz = dz1 ∧ · · · ∧ dzn by replacing dzi with d¯ zj (in the ith position). Then ddφf = 2Re

• n
• i,j=1

∂2f ∂¯ zi∂¯ zj Zij

• + (∆f)φ

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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

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. Case 2. G ≡ LAG = the Lagrangian n-planes .

tion of f to every minimal Lagrangian submanifold is sub- harmonic. NOTE: ddc 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 p-plurisubharmonic functions PSH(X, p) p-convexity A p-analogue of the Monge-Amp` ere operator.

f ∈ PSH(X, p) ⇔

• f is subharmonic on all

p dim′l minimal submanfolds

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, L1

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

CONVEXITY Suppose X is a non-compact.

• DEFINITION. If K ⊂⊂ X, the G-convex hull of K is
• K ≡ {x ∈ X : f(x) ≤ sup

K

f for all f ∈ PSH(X)}

1) If K ⊂⊂ X, then K ⊂⊂ X. 2) There exists a G-plurisubharmonic proper exhaustion function f on X. Such manifolds are called G-convex. 18

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. 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 For f ∈ PSH(X), define W(f) ≡ {x ∈ X : f is partially pluriharmonic at x}.

Core(X) ≡

• f∈P SH(X)

W(f)

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

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 trξHessρ ≥ 0 for all ξ ∈ G tangent to ∂Ω If trξHessρ > 0, all such ξ, ∂Ω is called strictly G-convex. This is independent of the choice of defining function ρ.

• f ∂Ω with respect to the outward pointing normal. Then

∂Ω is G-convex if and only if tr

• B
• ξ
• ≤ 0

for all G-planes ξ which are tangent to ∂Ω. 21

∂Ω strictly G-convex ⇒ Ω is strictly G-convex at ∞. Let δ = −ρ be the associated “distance function”. Then f ≡ −log δ is strictly G-psh outside a compact subset of Ω.

Ω strictly φ-convex at ∞ ⇒ ∂Ω is φ-convex. There is a partial converse for calibrations φ.

If −logδ is G(φ)-psh near ∂Ω, then ∂Ω is G(φ)-convex. 22

THE LEVI PROBLEM. Drop strictness.

• OBLEM. For which geometries is it true that :

∂Ω is G-convex ⇔ Ω is G-convex at infinity?

23

G-FREE SUBMANIFOLDS

no p-planes ξ ∈ G tangential to M. Every submanifold of dimension < p is G-free. If G = G(p, X), generic p-dim’l submanifolds are G-free. Suppose M ⊂ X is a closed submanifold and set fM(x) ≡ dist(x, M)2

strictly G-plurisubharmonic at each point in M (and hence in a neighborhood of M). 24

The existence of G-free submanifolds insures the existence

• f many strictly G-convex domains in X.

there exists a fundamental neighborhood system F(M) of M consisting of strictly G-convex neighborhoods. Moreover, (a) M is a deformation retract of each U ∈ F(M). (b) PSH(V, φ) is dense in PSH(U) if U ⊂ V and V, U ∈ F(M).

• EXAMPLE. Let M ⊂ X be any submanifold of dimension < p.

Then M has a fundamental system of neighborhoods each strictly G-convex and homotopy equivalent to M.

ibrated geometries. Suppose for instance that X is Calabi- Yau with special Lagrangian calibration. Then any com- plex submanifold Y ⊂ X is SL-free. It follows that any smooth submanifold of Y is also SL-free. 25

The following two classes of subsets of X. (1) Closed subsets of G-free submanifolds. (2) Zero sets of non-negative strictly G-psh functions. are basically the same. 26

MORSE THEORY. AND THE TOPOLOGY OF G-CONVEX SPACES. In many cases Morse Theory implies that there is an integer q = q(n) such that Any strictly G-convex manifold of dimension n has the homotopy-type of a complex of dimension ≤ q(n). The theorem above shows there are many G-convex manifolds of dimension = q(n).

Stein manifolds. 27

THE DIRICHLET PROBLEM FOR THE GENERALIZED MONGE-AMP` ERE EQUATION Let Ω ⊂⊂ X be open with smooth boundary ∂Ω. Suppose Ω is strictly G-convex, i.e., there is a defining function for Ω which is strictly G-plurisubharmonic on Ω.

a function F ∈ C(Ω) ∩ PSH(Ω) which satisfies: (a) F is partially pluriharmonic on Ω, and (b) F

• ∂Ω = u.

F is the upper-semi-continuous regularization of f(x) ≡ sup

g∈F(u)

g(x) F(u) ≡ {g ∈ PSH(Ω) : g ≤ u on ∂Ω}.

ferential operator M (generalized Monge-Amp` ere operator) with the property that if F ∈ C2(Ω) ∩ PSH(Ω), then F is partially pluriharmonic if and only if M(F) = 0 28

POSITIVITY. (X, φ) a calibrated manifold A p-dimensional current T (linear functional on p-forms)

• n X is represented by integration if there exist

(i) A Radon measure T, and (ii) T-measurable field of unit p-vectors − → T , so that T(α) =

• α(−

→ Tx)dT(x)

− → Tx ∈ Convex Hull of G(φ) for T − a.a.x

• THEOREM. φ-Positive currents are homologically mass mini-

mizing.

• THEOREM. φ-Positive currents which are rectifiable and d-

closed are positive integer linear combinations of φ-submanifolds with codimension-2 singularities. 29

GENERAL PHILOSOPHY. Our 1982 Acta paper introduced calibrations φ with

• φ-submanifolds, and more generally
• φ-positive currents

Our current work concerns the dual of these objects.

• Functions which morally lie in the polar cone

Both the φ-positive currents and the φ-plurisubharmonic functions exist in abundance. Their natural interaction is fruitful in understanding the ge-

• metry and analysis of calibrated manifolds

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SUPPORT OF φ-POSITIVE CURRENTS. Consider (X, φ) with ∇φ = 0. Let ∂φ∂ : D′

p(X) −

→ D′

0(X)

denote the formal adjoint of the operator ddφ.

positive current with compact support in X. If ∂φ∂T is ≤ 0 (a non-positive measure) on X − K, then supp T ⊂

• K ∪ Core(X)

. 31

SUPPORT OF φ-POSITIVE CURRENTS. Consider (X, φ) with ∇φ = 0. Let ∂φ∂ : D′

p(X) −

→ D′

0(X)

denote the formal adjoint of the operator ddφ.

positive current with compact support in X. If ∂φ∂T is ≤ 0 (a non-positive measure) on X − K, then supp T ⊂

• K ∪ Core(X)

.

• Y. If ∂φ∂T ≤ 0 on X, then

supp T ⊂ Core(X).

• Y. If (X, φ) is strictly G(φ)-convex and K =

K. Then supp{∂φ∂T} ⊂ K ⇒ suppT ⊂ K. In particular, there are no φ-positive currents which are com- pactly supported without boundary on X. . 32

DUVAL-SIBONY DUALITY. Suppose (X, φ) is strictly G(φ)-convex, and fix a com- pact subset K ⊂ X

K if and only if there exists a φ-positive current T with compact support and a probability measure µ on K such that ∂φ∂ T = µ − [x] (†) T is a Green’s current for (K, x), µ is a Poisson-Jensen measure for (K, x), and equation (†) is the Poisson-Jensen equation. 33