Weighted Bergman spaces of domains with Levi-flat boundary: two case studies

Weighted Bergman spaces of domains with Levi-flat boundary: two case studies

(arXiv:1703.08165)

Masanori Adachi (足立 真訓)

Tokyo University of Science

March 25, 2017

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Introduction: toy case

X: complex manifold, Ω⊂X: domain.

General problem (generalized Levi’s problem)

What kinds of geometry of ∂Ω control O(Ω)? Study their relation quantitatively.

Toy case:

CP¹ =C∪ {∞}. C, D={z ∈C| |z|<1}.

dimO(C) =∞. dimO(D) =∞.

dimO ∩L^∞(C) = 1̸= dimO ∩L^∞(D) =∞.

(∵) Liouville. / Compactification. Riemann’s removable singularity theorem + maximum principle.

(∵) D⋐C. Obtain bounded holomorphic functions on D by restricting those on C.

∂C={∞}, ∂D=S¹.

Strict pseudoconvexity and Levi-flatness

(X,J): complex manifold, Ω⊂X: domain. M =∂Ω: smooth real hypersurface.

Definition (Strictly pseudoconvex and Levi-flat)

T_M^1,0≃T_M∩JT_M ⊂T_M is called the Levi distribution.

M is sψc :⇐⇒ T_M^1,0 defines a positive contact structure.

M is Levi-flat :⇐⇒ T_M^1,0 defines a foliation (Levi foliation).

Example: X =Cⁿ, n≥2.

Ω ={(z^′,zn)∈Cⁿ|Imzn>∥z^′∥²}: sψc.

Ω ={(z^′,zn)∈Cⁿ|Imzn>0}: Levi-flat. M =⊔

t∈RCⁿ⁻¹× {t}.

Grauert:

Ω⋐X, M: sψc =⇒ dimO(Ω) =∞. Ω a proper modification of a Stein space, and dimO ∩L^∞(Ω) =∞.

∃Ω⋐X s.t. M: Levi-flat and dimO(Ω) = 1(only constant!).

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Holomorphic disk bundles and flat circle bundles

Σ: closed Riemann surface of genus ≥2. Fix its uniformization Σ≃D/π₁(Σ). Take a representation ρ:π₁(Σ)→Aut(D)⊂Aut(CP¹), Diffeo⁺(S¹).

Definition (suspension)

Xρ:= Σ×ρCP¹ :=D×CP¹/(z,w)∼(γz, ρ(γ)w) for γ ∈π1(Σ).

Ω_ρ:= Σ×ρD :=D×D/π₁(Σ).

M_ρ:= Σ×ρS¹ :=D×S¹/π₁(Σ).

We regard X_ρ, Ω_ρ, M_ρ as CP¹-bundle, D-bundle, S¹-bundle over Σ respectively by first projection. They are flat bundles in the sense that horizontal disks D× {t} (t∈CP¹) give a holomorphic foliation by Riemann surfaces on X_ρ and preserve Ω_ρ andM_ρ. In particular, Mρ is a Levi-flat real hypersurface.

Theorem of Grauert, Diederich–Ohsawa

Σ: closed Riem. surface of g ≥2, ρ:π1(Σ)→Aut(D): rep.

new representations called conjugation are given by π₁(Σ)→Aut(D), γ 7→α◦ρ(γ)◦α⁻¹ for each α∈Aut(D).

Xρ, Ωρ, Mρ do not change up to conjugations.

Theorem (Grauert, Diederich–Ohsawa)

1 ρ is not to conjugate to rotations

=⇒ Ω_ρ is a proper modification of a Stein space.

2 ρ is conjugate to rational rotations

=⇒ Ωρ is holomorphically convex, but not Stein.

3 ρ is conjugate to rotations including an irrational rotation

=⇒ dimO(Ω_ρ) = 1.

Examples

1 ρ(γ) =γ (Deck transformation),[ρ]∈ T(Σ) (Teichm¨uller space).

2 ρ(γ)≡id_D =⇒ Ωρ= Σ×D. O(Ωρ) =O(D).

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The non-existence conjecture of Levi-flats in CP

²

Conjecture

1 Let Fe be a singular codimension one holomorphic foliation on CP². Then, for any leaf L of Fe, L∩Sing(Fe)̸=∅.

2 There is no smooth closed Levi-flat real hypersurface in CP². (Poincar´e–Bendixson) Let Fe be a flow on RP². Then, for any flow line L of Fe, L∩Sing(F)e ̸=∅, orL contains a periodic orbit.

(Cerveau) Suppose we have a counterexample Fe. Then, the non-trivial minimal set M of Fe is either having very restricted holonomy, or being real-analytic Levi-flat.

On the other hand, an real-analytic Levi-flat in CP² produces a counterexample Fe by analytic continuation.

(Lins Neto, Siu) The conjecture is true for CP^≥3. The main point of the proof is Takeuchi’s theorem: the complement of hypothetical minimal set or Levi-flat is Stein.

Holomorphic normal bundle

Ω⋐X, ∂Ω =M: smooth Levi-flat. F: Levi foliation Definition (holomorphic normal bundle)

N_M^1,0:= (T_X^1,0|M)/T_M^1,0≈C⊗TM/T_F.

(Brunella, Ohsawa, Biard–Iordan) There is no smooth closed Levi-flat M in compact K¨ahler manifolds of dimension ≥3 having positive N_M^1,0 (i.e. N_M^1,0 has positive curvature along F).

N_M^1,0 >0 =⇒ Ω is a proper modification of a Stein space.

For circle bundles, N_M^1,0

ρ ≥0 if ρ is not conjugate to rotations.

In particular, N_M^1,0

ρ >0 if [ρ]∈ T(Σ).

The argument for the non-existence in CP^≥3 breaks down for CP². We need to find some other quantitative approach, and we try through generalized Levi’s problem.

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General Problem

Σ: closed Riem. surface of g ≥2, ρ:π1(Σ)→Aut(D): rep.

X_ρ⊃Ω_ρ, M_ρ, Ω^′_ρ:=X_ρ\Ω_ρ. Fρ: Levi foliation. N_M^1,0

ρ: hol. n. bdle.

Probem

Study the dependency on ρ of spaces of holomorphic functions on Ω_ρ and Ω^′_ρ and CR functions on M_ρ with growth conditions. In particular, study its mechanism from the viewpoint of ergodicity of Fρ and geometry of N_M^1,0

ρ.

(A.–Brinkschulte) Curvature restriction of smooth closed Levi-flats in CP² was improved by studying infinite

dimensionality of the Bergman space of the canonical bundle over its complement.

Today: we study the case ρ(γ) =γ (Deck transformation).

Bergman space

Ω⋐X, M =∂Ω: smooth. dV: volume form on X. δ :X →R, Ω ={δ >0}, dδ ̸= 0 on ∂Ω. α≥ −1.

Definition (Weighted Bergman space, Hardy space)

A²_α(Ω) :={f ∈ O(Ω)| ⟨f,f⟩α<∞},

⟨f,g⟩α :=

{∫

Ωf gδ^αdV/Γ(α+ 1) for α >−1, lim_α↘−1⟨f,g⟩α for α=−1.

The Hardy space (α =−1) is the space of holomorphic functions which have L² boundary value. The boundary values are CR functions on M.

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CR functions on Levi-flats

(M,F): compact Lev-flat CR manifold, i.e.,

M: compact real manifold, F: real codimension one non-singular foliation by complex manifolds.

Definition (CR function)

A CR function is a function which are holomophic along F.

Transverse regularity are not guaranteed.

(Inaba) Any continuous CR function on M are leafwise

constant. In particular, continuous CR functions are constant if F has a dense leaf.

(Ohsawa–Sibony, A.) The finite/infinite dimensionality of the space of CR sections of a fixed smooth CR line bundle depend on transverse regularity we require.

Liouvilleness

Σ: closed Riemann surface of genus ≥2, ρ0:π1(Σ)→Aut(D), γ7→γ (Deck transformation).

X :=X_ρ0, Ω := Ω_ρ0, M :=M_ρ0, Ω^′ :=X_ρ0\Ω_ρ0. Corollary of E. Hopf’s ergodicity theorem

Ω, Ω^′ are Liouville, i.e., dimO ∩L^∞(Ω) = dimO ∩L^∞(Ω^′) = 1.

Proof

Let f be a bounded holomorphic function on Ω or Ω^′. Consider the boundary value of f, which is leafwise holomorphic.

M is diffeomorphic to the unit tangent bundle of Σ, and the Levi foliation is isomorphic to the unstable foliation of geodesic flows of Σ w.r.t. the Poincar´e metric.

The ergodicity of geodesic flow w.r.t. the Lebesgue measure implies that any bounded leafwise harmonic function is a.e.

constant. Hence, f is constant.

Another proof based on a property of 3F2 will be given later.

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Hardy space and weighted Bergman spaces of Ω

O(Ω)≃ {f ∈ O(D×D)|f(z,w) =f(γz, γw), γ∈π1(Σ)}.

O(Ω^′)≃ {f ∈ O(D×D)|f(z,w) =f(γz, γw), γ∈π₁(Σ)}.

Refinement of E. Hopf’s ergodicity theorem by L. Garnett gives dimA²₋₁(Ω) = dimA²₋₁(Ω^′) = 1.

Corollary of the technique of Berndtsson–Charpentier

dimA²_α(Ω) = dimA²_α(Ω^′) =∞, ∀α >−1/2.

The 1/2 is the Diederich–Fornaess index of Ωand Ω^′, which is the supremum of H¨older exponents of non-trivial (s)psh functions having zero boundary values.

(Fu–Shaw, A.–Brinkschulte) The 1/2 is the best possible as the Diederich–Fornaess index.

(A.) The 1/2 corresponds to the ergodicity of geodesic flow w.r.t. the Lebesgue measure in a sense.

Main result

Main result (A., arXiv:1703.08165 + in preparation.)

∃I:

⊕∞ n=0

H⁰(Σ,K_Σ^⊗n),→ O(Ω), ∃I^′:

⊕∞ n=0

Ker(∆−λ_nI),→ O(Ω^′) where ∆is the Laplace–Beltrami operator of Σ w.r.t. Poincar´e metric, and

H⁰(Σ,K_Σ^⊗n) :={holomorphic n-differential onΣψ=ψ(τ)(dτ)^⊗n}, Ker(∆−λnI) :={f : Σ→C|∆f =λnf}.

The images of I and I^′ are dense in compact open topology, and, moreover, contained in A²_α(Ω)and A²_α(Ω^′) resp. for∀α >−1.

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Construction of I and I

^′

Ω contains a divisor D ={(z,z)|z ∈D}/π₁(Σ)≃Σ.

Ω^′ contains a totally real D^′ :={(z,z)|z ∈D}/π₁(Σ)≈Σ, and Ω^′ is characterized as the Grauert tube of maximal radius.

I: ⊕_∞

n=0H⁰(Σ,K_Σ^⊗n),→ O(Ω)is given by I(ψ)(z,w) :=

∫ _w

z

1 B(n,n)

((w −τ)(τ−z) (w −z)dτ

)_⊗(n−1)

ψ(τ)(dτ)^⊗n for ψ∈H⁰(Σ,K_Σ^⊗n), n ≥1, where we write ψ=ψ(τ)(dτ)^⊗n on the uniformizing coordinate τ ∈D. The well-definedness follows from a property of cross ratios.

I^′: ⊕_∞

n=0Ker(∆−λnI),→ O(Ω^′) is given by the analytic continuation of f ∈Ker(∆−λnI) regarded as a real-analytic function on D^′ to D×D. The well-definedness follows from a known fact on PDE.

Outline of proof of integrability

I is obtained by optimal L²-jet extension from D toΩ. We may regard ψ∈H⁰(Σ,K_Σ^⊗n) as an n-th order jet of holo func along D via

K_Σ ≃T_Σ^∗ ≃T_D^∗ ≃N_D/Ω^∗ , D={(z,z)|z ∈D}/π₁(Σ)⊂Ω, Step 1. We work on a non-holomorphic coordinate of Dz×Dw given by (z,t), t := (w−z)(1−zw)⁻¹. Expand f =f(z,w)∈ O(Ω)^π1(Σ) as f =∑_∞

n=0fn(z)tⁿ, then {fn} enjoys

∂f_n

∂z + nz

1− |z|²f_n+ n−1

1− |z|²f_n−1= 0.

Put φ_n:=f_n(z) (√

2dz 1−|z|²

)_⊗n

∈C^(0,0)(Σ,K_Σⁿ). Then {φ_n} satisfy

∂φ₀ = 0, ∂φ_n=−n−1

√2 φ_n−1⊗ω (n≥1) where ω= 2dz⊗d z/(1− |z|²)².

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Outline of proof of integrability – continued

Step 2. Let ψ∈H⁰(Σ,K_Σ^⊗N). Putφ_n:= 0 for n<N and φ_N :=ψ.

We pick the L² minimal solution to

∂φ_n=−n−1

√2 φ_n−1⊗ω

inductively and determine φn forn >N. The spectral decomposition of the complex laplacian tells us the L² minimal solutions are

φ_N+m=∂^∗_N+mG_N+m⁽¹⁾ (

−N+m−1

√2 φ_N+m−1⊗ω )

=−

√2(N+m−1)

m(2N+m−1)∂^∗_N+m(φN+m−1⊗ω)

where ∂^∗_n is the formal adjoint of ∂:C^(0,0)(Σ,K_Σ^⊗n)→C^(0,1)(Σ,K_Σ^⊗n) and Gn⁽¹⁾ is the Green operator on C^(0,1)(Σ,K_Σ^⊗n).

Outline of proof of integrability – continued

³

Step 3. For α >−1, the convergence of f =∑_∞

n=0f_n(z)tⁿ, φn=fn(z)

(√ 2dz 1−|z|²

)_⊗n

in L²_α(Ω)follows from

∥f∥²α=π

∑∞ n=0

∥φn∥² Γ(n+ 1) Γ(n+α+ 2)

=π

∑∞ m=0

∥φ_N+m∥² Γ(N+m+ 1) Γ(N+m+α+ 2)

=π∥ψ∥²∑^∞

m=0

Γ(N+m+ 1) Γ(N+m+α+ 2)

(2N−1)!

{(N−1)!}²

{(N+m−1)!}² m!(2N+m−1)!

=π∥ψ∥² Γ(N+ 1) Γ(N+ 2 +α)³F₂

( N+ 1,N,N 2N,N+ 2 +α; 1

)

<∞

where we used δ := 1−₁^w₋⁻_zw^z2, dV = _|1−zw⁴ _|4 i

2dz∧d z∧₂ⁱdw∧d w. Similar computation shows f ∈ O(Ω), and the Liouvilleness of Ω.

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Outline of proof of integrability – continued

⁴

Step 4. Want to show

∑∞ n=0

f_n(z)tⁿ=

∫ _w

z

1 B(N,N)

((w−τ)(τ −z) (w−z)dτ

)_⊗(N−1)

ψ(τ)(dτ)^⊗N. Enough to show the desired equality on {0} ×D.

∑∞ n=0

fn(0)tⁿ= (2N−1)!

(N−1)!

∑∞ m=0

(N+m−1)!

(2N+m−1)!

1 m!

∂^mψ

∂z^m(0)t^N+m

= (2N−1)!

(N−1)!t^N

∫ ₁

0

dtN. . .

∫ _t3

0

dt2

∫ _t2

0

t₁^N−1ψ(tt1)dt1

= (2N−1)!

(N−1)!t^N

∫ ₁

0

t₁^N−1(1−t₁)^N−1

(N−1)! ψ(tt₁)dt₁

=

∫ _t

0

1 B(N,N)

((t−τ)τ t

)_(N−1)

ψ(τ)dτ.

A Forelli–Rudin construction

The reproducing kernel Bα of A²_α is called the weighted Bergman kernel. Bα(z,w) =∑_∞

j=1ej(z)ej(w) for any orthonormal basis of A²_α {e₁,e₂, . . .}.

Corollary

The weighted Bergman kernel of Ω is B_α((z,w); (z^′,w^′)) = Γ(α+ 2)

π²(4g −4)+ 1

π

∑∞ n=1

1 c_n,α

1 B(n,n)²

∫

τ∈zw

∫

τ^′∈z^′w^′

Bn(τ, τ^′)(dτ ⊗dτ^′)^⊗n ([w, τ,z]⊗[w^′, τ^′,z^′])^⊗(n−1). where g is the genus of Σ, Bn(τ, τ^′)(dτ⊗dτ)^⊗n is the Bergman kernel of holomorphic n-differentials, i.e. of H⁰(Σ,K_Σ^⊗n), and

cn,α= Γ(n+ 1) Γ(n+ 2 +α)³F2

( n+ 1,n,n 2n,n+ 2 +α; 1

)

,[w, τ,z] = (w −z)dτ (w−τ)(τ −z).

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