The Geometry of Grauert Tubes and Complexification of Symmetric Spaces

全 Grauert 管上的零熵

Zero entropy on entire Grauert tubes

本文证明具有全 Grauert 管的解析流形其测地流具有零拓扑熵，并由此给出 3-流形的完整分类及若干 4、5-流形的拓扑限制与反例。

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…

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 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 …

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。

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。

Invariant envelopes of holomorphy in the complexification of a Hermitian symmetric space

In this paper we investigate invariant domains in $\, Ξ^+$, a distinguished $\,G$-invariant, Stein domain in the complexification of an irreducible Hermitian symmetric space $\,G/K$. The domain $\,Ξ^+$, recently introduced by Krötz and Opdam, contains the crown domain $\,Ξ\,$ and it is maximal with respect to properness of the $\,G$-action. In the tube case, it also contains $\,S^+$, an invariant Stein domain arising from the compactly causal structure of a symmetric orbit in the boundary of $\,Ξ$. We prove that the envelope of holomorphy of an invariant domain in $\,Ξ^+$, which is contained n…

Invariant Stein domains in Stein symmetric spaces and a non-linear complex convexity theorem

We prove a complex version of Kostant's non-linear convexity theorem. Applications to the construction of G-invariant Grauert tubes of non-compact Riemannian symmetric G/K spaces are given.