Existence of nonconstant CR-holomorphic functions of polynomial growth in Sasakian manifolds

Let ( M , J , θ ) be a complete noncompact Sasakian ( 2 ⁢ n + 1 ) -manifold.

1. A 𝑝-form 𝛾 is called basic if i ⁢ ( T ) ⁢ γ = 0 and L T ⁢ γ = 0 .

2. Let Λ B p be the sheaf of germs of basic 𝑝-forms and let Ω B p be the set of all global sections of Λ B p .

Lemma A.1 (CR Bochner–Kodaira–Nakano identity)

Suppose that ( M , T 1 , 0 ⁢ ( M ) , θ ) is a strictly pseudoconvex Sasakian ( 2 ⁢ n + 1 ) -manifold and 𝐷 is the Tanaka connection in a CR-holomorphic vector bundle 𝐸 over 𝑀. Then the complex sub-Laplace operators Δ ′ and Δ ∂ ̄ B acting on 𝐸-valued basic forms satisfy the identity

where Δ ∂ ̄ B = ∂ ̄ B ∗ ⁢ ∂ ̄ B + ∂ ̄ B ⁢ ∂ ̄ B ∗ . Here Θ ⁢ ( E ) := D 2 ⁢ s is the curvature operator.

Proof

Let 𝐿 be the Lefschetz operator defined by L ⁢ s = 1 2 ⁢ d ⁢ θ ∧ s and Λ = L ∗ its adjoint operator. It follows from [46] (see also [20]) that the following transverse Kähler identities hold:

Hence we have

On the other hand, by the Jacobi identity, we have

where we use [ D ′ , ∂ ̄ B ] = D 2 = Θ ⁢ ( E ) . Then we derive the CR analogue of the Bochner–Kodaira–Nakano identity. ∎

Let ( M , J , θ ) be a complete noncompact Sasakian ( 2 ⁢ n + 1 ) -manifold of nonnegative pseudohermitian Ricci curvature with 0 ≤ u ⁢ ( x ) ≤ exp ⁡ ( a ⁢ r ⁢ ( x ) + b ) for some a , b > 0 , u ∈ C c ∞ ⁢ ( M ) . We denote

on M × [ 0 , ∞ ) with the CR heat kernel H ⁢ ( x , y , t ) . Then, for any positive numbers ϵ > 0 and T > 0 , there exist C 1 ⁢ ( n , ϵ ) > 0 and C 2 ⁢ ( n , a , b , ϵ , T ) > 0 such that

Here ( x , t ) ∈ r − 1 ⁢ ( [ T , + ∞ ) ) × [ 0 , T ] , r ⁢ ( x ) = d cc ⁢ ( x , o ) is the Carnot–Carathéodory distance d cc between 𝑥 and 𝑜 for some fixed point o ∈ M and B ⁢ ( x , r ) is the ball with respect to d cc .

Proof

If s ≥ ϵ ⁢ r ⁢ ( x ) , then

where we use the fact that

Here V x ⁢ ( s ) and | u ̂ | B x ⁢ ( s ) denote the volume of the Carnot–Carathéodory ball B ⁢ ( x , s ) = B x ⁢ ( s ) with respect to the volume element d ⁢ μ = θ ∧ ( d ⁢ θ ) n and the average of the absolute value of the function 𝑢 over B x ⁢ ( s ) respectively. Due to r ⁢ ( x ) ≥ T , we have

It follows from (B.1), (B.2) and

for s ∈ [ 0 , + ∞ ) that we are able to derive

by the CR heat kernel estimate and the CR volume doubling property [2]. This implies that

On the other hand, it is not difficult to deduce that

The proof is accomplished. ∎

Let ( M , J , θ ) be a complete noncompact Sasakian ( 2 ⁢ n + 1 ) -manifold of nonnegative pseudohermitian bisectional curvature and let v ⁢ ( x , t ) be a nonnegative solution to the CR heat equation on M × [ 0 , T ] . Then ∥ η α ⁢ β ̄ ⁢ ( x , t ) ∥ is a subsolution to the CR heat equation, where η α ⁢ β ̄ = v α ⁢ β ̄ + v β ̄ ⁢ α . Furthermore, if

on M × [ 0 , T ] .

It is not difficult to observe that the existence of the function h ⁢ ( x , t ) is ensured by (B.3).

Proof

By [6, (3.5)] and the vanishing pseudohermitian torsion, we see that

Therefore,

By straightforward calculation, we have

That is to say that

It is clear that ∥ η α ⁢ β ̄ ⁢ ( x , t ) ∥ − h ⁢ ( x , t ) is a subsolution to the CR heat equation. It follows from (B.4) and [3, Lemma 4.5] that ∥ η α ⁢ β ̄ ⁢ ( x , t ) ∥ ≤ h ⁢ ( x , t ) . ∎

Let ( M , J , θ ) be a complete noncompact Sasakian ( 2 ⁢ n + 1 ) -manifold of nonnegative pseudohermitian Ricci curvature with

for any positive number 𝑎. Here 𝜂 is chosen as in Lemma B.2. Then there is a positive function τ ⁢ ( R ) with τ ⁢ ( R ) → 0 + as 𝑅 goes to infinity such that, for any positive number 𝑇,

on A ⁢ ( o ; R 2 , R ) × [ 0 , T ] . Here h ⁢ ( x , t ) is chosen as in (B.5).

Proof

Fix T > 0 and let 𝑅 ( ≥ T ) be a sufficiently large positive number (we may assume supp ⁡ ( u ) ⊆ B ⁢ ( o , R 8 ) ). Let x ∈ A ⁢ ( o ; R 2 , R ) . We have, for a sufficiently small a > 0 ,

Here we utilize the CR heat kernel estimate, the facts that d cc ⁢ ( x , y ) ≥ R 2 16 and the CR volume doubling property V o ⁢ ( R ) ≤ V x ⁢ ( 3 ⁢ R ) ≤ C ⁢ ( n ) ⁢ V x ⁢ ( R ) . ∎

Let ( M , J , θ ) be a complete noncompact Sasakian ( 2 ⁢ n + 1 ) -manifold of nonnegative pseudohermitian Ricci curvature and let v ⁢ ( x , t ) be the solution to the CR heat equation with the initial condition

u 0 = 0 and η α ⁢ β ̄ = v α ⁢ β ̄ + v β ̄ ⁢ α . Then

Proof

Because

and u ∈ C c ∞ ⁢ ( M ) , we see that | v ⁢ ( x , t ) | ≤ C in M × [ 0 , + ∞ ) . By the fact that

and with the help of the cut-off function φ ∈ C c ∞ ⁢ ( M ) satisfying 0 ≤ φ ≤ 1 , φ | B o ⁢ ( r ) = 1 , φ | M \ B o ⁢ ( 2 ⁢ r ) = 0 and | Δ b ⁢ φ ⁢ ( x ) | ≤ 1 r 2 ⁢ ( x ) for any positive number 𝑟, it is clear that, for any r ≥ 1 ,

From the CR Bôchner formula and u 0 = 0 (this implies that v 0 = 0 ), we obtain

With a method similar to the preceding, we have, for any r ≥ 1 ,

This enables us to deduce that

We could drop the assumption of u 0 = 0 by estimating the integral

by (B.6).

Proof

We will apply the maximum principle to confirm the validity of this theorem. The detail is as below. For any small positive number ϵ > 0 , define

where

We observe that η ̃ α ⁢ β ̄ ⁢ ( x , 0 ) > 0 for any x ∈ M . By Lemma B.2 and Lemma B.3, η ̃ α ⁢ β ̄ ⁢ ( x , t ) > 0 on ∂ B o ⁢ ( R ) × [ 0 , T ] for any sufficiently large number R > 0 . Suppose that there is a point ( x 0 , t 0 ) ∈ B o ⁢ ( R ) × ( 0 , T ] such that η ̃ α ⁢ β ̄ ⁢ ( x 0 , t 0 ) < 0 . There is a number t 1 ∈ [ 0 , t 0 ) such that η ̃ α ⁢ β ̄ ⁢ ( x , t ) ≥ 0 on B o ⁢ ( R ) × ( 0 , t 1 ] and the bottom spectrum of η ̃ α ⁢ β ̄ ⁢ ( x 1 , t 1 ) is

for some x 1 ∈ B o ⁢ ( R ) . We assume that η ̃ α ⁢ β ̄ ⁢ ( x 1 , t 1 ) is a diagonal matrix and η ̃ γ ⁢ γ ̄ ⁢ ( x 1 , t 1 ) = 0 for γ ∈ T x 1 1 , 0 ⁢ ( M ) with ∥ γ ∥ = 1 . Here the vector field 𝛾 is chosen as one of the bases of the CR structure T x 1 1 , 0 ⁢ ( M ) . Therefore, we have