Symplectic geometry and the uniqueness of Grauert tubes

Nonvanishing of Cartan CR curvature on boundaries of Grauert tubes around hyperbolic surfaces

We show that the boundaries of thin strongly pseudoconvex Grauert tubes, with respect to the Guillemin-Stenzel Kähler metric canonically associated with the Poincaré metric on closed hyperbolic real-analytic surfaces, has nowhere vanishing Cartan CR-curvature. This result provides a wealth of examples of compact $3$-dimensional Levi nondegenerate CR manifolds having no CR-umbilical point. We provide two proofs utilizing two recent formulas for determining the Cartan CR-curvature of any local $\mathcal{C}^6$-smooth hypersurfaces in $\mathbb{C}^2$. One was obtained in 2012 by the second named au…

Scaling asymptotics for Szegő kernels on Grauert tubes

Let $M_τ$ be the Grauert tube of radius $τ$ of a closed, real analytic manifold $M$. Associated to the Grauert tube boundary is the orthogonal projection $Π_τ\colon L^2(\partial M_τ) \to H^2(\partial M_τ)$, called the Szegő projector. Let $D_{\sqrtρ}$ denote the Hamilton vector field of the Grauert tube function $\sqrtρ$ acting as a differential operator. We prove scaling asymptotics for the spectral localization kernel of the Toeplitz operator $Π_τD_{\sqrtρ} Π_τ$. We also prove scaling asymptotics for the tempered spectral projections kernel $P_{χ, λ}(z,w) = \sum_{λ_j \le λ} e^{-2τλ_j} ϕ_{λ_j…

On rigidity of Grauert tubes over homogeneous Riemannian manifolds

Given a real-analytic Riemannian manifold $X$ there is a canonical complex structure, which is compatible with the canonical complex structure on $T^*X$ and makes the leaves of the Riemannian foliation on $TX$ into holomorphic curves, on its tangent bundle. A {\it Grauert tube} over $X$ of radius $r$, denoted as $T^rX$, is the collection of tangent vectors of $X$ of length less than $r$ equipped with this canonical complex structure. In this article, we prove the following two rigidity property of Grauert tubes. First, for any real-analytic Riemannian manifold such that $r_{max}>0$, we show th…

Poisson 和 Szegö 核在 Grauert 管边界上的标度渐近（继 Zelditch、Chang 和 Rabinowitz 之后）

Poisson and Szegö kernel scaling asymptotics on Grauert tube boundaries (after Zelditch, Chang and Rabinowitz)

本文回顾并扩展了 Grauert 管边界上 Poisson 和 Szegö 核的标度渐近研究，给出了近对角情形的首阶项显式描述及多项式次数增长估计，并允许在更广泛的变量范围内进行重标度。

On the algebraicization of certain Stein manifolds

To every real analytic Riemannian manifold M there is associated a complex structure on a neighborhood of the zero section in the real tangent bundle of M. This structure can be uniquely specified in several ways, and is referred to as a Grauert tube. We say that a Grauert tube is entire if the complex structure can be extended to the entire tangent bundle. We prove here that the complex manifold given by an entire Grauert tube is, in a canonical way, an affine algebraic variety. In the special case M = the 2-sphere, we show that any entire Grauert tube associated to a metric (not necessarily …

Grauert管边界上Poisson核与Szegő核的等变渐近展开

Equivariant scaling asymptotics for Poisson and Szegő kernels on Grauert tube boundaries

本文研究紧致实解析黎曼流形上Grauert管边界处，等距群作用下复延拓特征函数与Hardy空间特征函数沿边界的等变渐近集中，并给出应用。

圆2球面的线丛与Grauert管Hardy量子化的交织

Intertwining the line bundle and Grauert-tube Hardy quantizations of the round 2-sphere

本文比较圆2球面单位余切丛上的两种Hardy量子化，计算其直线归一化重叠，推导相关迹级数与渐近，给出等变酉算子及Fredholm行列式等结果。

Complex zeros of real ergodic eigenfunctions

We determine the limit distribution (as $λ\to \infty$) of complex zeros for holomorphic continuations $ϕ_λ^{\C}$ to Grauert tubes of real eigenfunctions of the Laplacian on a real analytic compact Riemannian manifold $(M, g)$ with ergodic geodesic flow. If $\{ϕ_{j_k} \}$ is an ergodic sequence of eigenfunctions, we prove the weak limit formula $\frac{1}{λ_j} [Z_{ϕ_{j_k}^{\C}}] \to \frac{i}π \bar{\partial} {\partial} |ξ|_g$, where $ [Z_{ϕ_{j_k}^{\C}}]$ is the current of integration over the complex zeros and where $\bar{\partial}$ is with respect to the adapted complex structure of Lempert-Szök…

Solving the $d$ and $\bar\partial$-equation in thin tubes and applications to mappings

We construct a family of integral kernels for solving the \bar\partial equation with C^k and Holder estimates in thin tubes around totally real submanifolds in complex Eulidean spaces (theorems 1.1 and 3.1). Combining this with the proof of a theorem of Serre we solve the d-equation with estimates for holomorphic forms in such tubes (theorem 5.1). We apply these techniques and a method of Moser to approximate C^k-diffeomorphisms between totally real submanifolds in C^n in the C^k-topology by biholomorphic mappings in tubes, by unimodular and symplectic biholomorphic mappings, and by holomorphi…