全实映射与独立映射

A Uniqueness Theorem for Holomorphic Mappings in the Disk Sharing Totally Geodesic Hypersurfaces

In this paper, we prove a Second Main Theorem for holomorphic mappings in a disk whose image intersects some families of nonlinear hypersurfaces (totally geodesic hypersurfaces with respect to a meromorphic connection) in the complex projective space $\mathbb{P}^k$. This is a generalization of Cartan's Second Main Theorem. As a consequence, we establish a uniqueness theorem for holomorphic mappings which intersects $O(k^3)$ many totally geodesic hypersurfaces.

流形到C^N的全实浸入和独立映射的最优性

Optimality for totally real immersions and independent mappings of manifolds into C^N

最佳的目标尺寸确定全实浸入和独立映射到复杂的仿射空间。我们的论证与Forster给出的结果相似，但我们尽可能地使用了可定向流形，因此对于偶数维的可定向流形得到了改进的结果。这导致新的例子表明，已知的浸没和浸没尺寸的全纯映射从Stein流形到仿射空间是最好的可能。

细管中$d$和$\bar\partial$方程的求解及其在映射中的应用

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

我们构造了一族积分核来求解\bar \partial方程，其中C^k和Holder估计在复欧氏空间中围绕全实子流形的细管中（定理1.1和3.1）。结合这一证明定理的塞尔，我们解决了d-方程估计全纯形式在这样的管道（定理5.1）。我们应用这些技巧和Moser的方法，通过管中的双全纯映射、单模和辛双全纯映射以及C^n的全纯自同构，来逼近C^n中全实子流形之间的C ^k-拓扑中的C^k-全纯自同构。

Grassmannian闭SU（l，m）-轨道间的全纯映射

Holomorphic maps between closed SU(l,m)-orbits in Grassmannian

格拉斯曼流形中的$SU(\ell, m)$的轨道具有齐次CR结构。本文研究了Grassmannian流形中将$SU(\ell,m)$的一个闭轨道发送到$SU(\ell',m')$的一个闭轨道的光滑CR映射的芽.我们证明了如果两个轨道的Levi形式的签名差不是太大，那么映射可以分解成一个简单的形式，并且其中一个因子扩展到环境格拉斯曼到另一个格拉斯曼关于标准度量的全测地线嵌入。作为应用，我们给出了Grassmannian流形中的光滑CR映射将$SU(\ell,m)$的闭轨道延伸到$SU(\ell',m')$的闭轨道的一个充分条件，作为Grassmannian流形到另一Grassmannian流形的全测地嵌入.

Topological invariants and Holomorphic Mappings

Invariants for Riemann surfaces covered by the disc and for hyperbolic manifolds in general involving minimizing the measure of the image over the homotopy and homology classes of closed curves and maps of the $k$-sphere into the manifold are investigated. The invariants are monotonic under holomorphic mappings and strictly monotonic under certain circumstances. Applications to holomorphic maps of annular regions in $\mathbb{C}$ and tubular neighborhoods of compact totally real submanifolds in general in $\mathbb{C}^n$, $n \geq 2$, are given. The contractibility of a hyperbolic domain with con…

全非负格拉斯曼流形上Wronski映射的无零点扇区

Zero-free sector of the Wronski map on the totally nonnegative Grassmannian

本文受Shapiro--Shapiro猜想启发，将非负系数多项式无零点扇区的经典结果推广到全非负格拉斯曼流形上多项式的Wronskian行列式，证明其在扇区$|\text{arg}(z)| < \fracπ{n}$内无复零点且界紧。