Finsler流形上的Laplace比较定理和有界估计(英文)

本文研究了Finsler流形上的距离函数的Laplacian.利用指标引理和文献[4]中主要方法,获得了Ricci曲率有函数下界的Laplacian比较定理,改进了文献[6]和文献[7]的相关结果.     

Finsler流形间调和映射的一个刚性定理

把无焦点黎曼流形的概念推广到了Finsler流形中.通过在无焦点Finsler流形上构造凸函数,得到了Finsler流形间调和映射的一个刚性定理.     

3-对称Finsler流形（英文）

3-对称Finsler流形是3-对称黎曼流形的推广.本文给出了3-对称Finsler流形的定义,并将3-对称Finsler流形用齐性空间的形式表示.同时,本文还给出了在齐性空间上存在3-对称Finsler度量的条件,并讨论了3-对称Finsler流形与3-对称黎曼流形的关系.最后,本文给出了自然约化的3-对称Finsler流形的旗曲率和曲率张量.     

双挠积复Finsler流形的局部对偶平坦性

设(M₁,F₁)和(M₂,F₂)是两个强凸的复Finsler流形,λ₁和λ₂是乘积流形M=M₁×M₂上的光滑实值函数,双挠积复Finsler流形(M₁×(λ₁,λ₂) M2,F)是在乘积流形上赋予了复Finsler度量F²=λ₁²F₁²+λ₂²F₂²的复Finsler流形.本文给出了双挠积复Finsler流形是局部对偶平坦流形的充要条件.     

Finsler流形中弧长第二变分与子流形(英文)

本文研究了Finsler流形中的子流形的相关问题.利用文[23,24]中引入的Finsler流形中的切曲率和法曲率的概念,计算出Finsler流形中测地线的一个新的第二变分公式,获得了关于Finsler子流形中几何不变量和拓扑不变量的一些新的关系,推广了文[4]的许多结果.     

Finsler流形上的Laplace算子

该文对Finsler流形上的微分式定义了整体内积，进而引入δ算子和Laplace算子。该文还给出了δ算子的局部坐标表达式并且证明了Laplace算子可以看成是Riemann流形上Laplace算子在Finsler流形上的扩张。     

Finsler流形间调和映射的第2变分

给出了Finsler流形间非蜕化映射的能量泛函的第1和第2变分公式, 并利用变分公式得到了从Finsler流形到Riemann流形的非常值稳定调和映射的若干不存在性定理.     

关于Finsler流形的曲率与基本群

Finsler流形是具备没有二次型限制的Riemann度量的微分流形,它是Riemann流形的最自然推广.本文概述Finsler流形的曲率与基本群方面的若干进展和新近结果,内容包含基本群的增长、流形和基本群的熵、第一Betti数和基本群的有限性定理等,为进一步发展整体Finsler几何抛砖引玉.     

稳定调和映照的不存在性

讨论了从Finsler流形到Riemannian流形的稳定调和映照的不存在性,得到了一个拼挤定理.     

Riemann-Finsler几何中的若干问题

Finsler几何是比Riemann几何更一般的微分几何,近几十年来取得了全新的实质性进展.本文就若干尚未解决的整体Finsler几何问题作一概要的综述,主要涉及Finsler子流形几何、Finsler流形的曲率和拓扑,以及Finsler-Einstein度量.     

Comparison theorems in Finsler geometry and their applications

We prove Hessian comparison theorems, Laplacian comparison theorems and volume comparison theorems for Finsler manifolds under various curvature conditions. As applications, we derive McKean type theorems for the first eigenvalue of Finsler manifolds, as well as generalize to Finsler manifolds a result on fundamental groups due to Milnor.The research of the Y. L. Xin was partially supported by NSFC of and SFECC.     

Eigenvalue control for a Finsler–Laplace operator

Using the definition of a Finsler–Laplacian given by the first author, we show that two bi-Lipschitz Finsler metrics have a controlled spectrum. We deduce from that several generalizations of Riemannian results. In particular, we show that the spectrum on Finsler surfaces is controlled above by a constant depending on the topology of the surface and on the quasireversibility constant of the metric. In contrast to Riemannian geometry, we then give examples of highly non-reversible metrics on surfaces with arbitrarily large first eigenvalue.     

Finsler空间中的Laplace算子及其性质

在Finsler空间中给出了一种非线性的Laplace算子Δ,得到了Laplace算子Δ满足的性质,同时指出了Δ与Riemann空间中Laplace算子的异同.     

Comparison theorems on Finsler manifolds with weighted Ricci curvature bounded below

We obtain the Laplacian comparison theorem and the Bishop-Gromov comparison theorem on a Finsler manifold with the weighted Ricci curvature Ric_∞ bounded below. As applications, we prove that if the weighted Ricci curvature Ric_∞ is bounded below by a positive number, then the manifold must have finite fundamental group, and must be compact if the distortion is also bounded. Moreover, we give the Calabi-Yau linear volume growth theorem on a Finsler manifold with nonnegative weighted Ricci curvature.     

Hodge theorem for the natural projection of complex horizontal Laplacian on complex Finsler manifolds

Let M be a compact complex manifold with a complex Finsler metric F. We define a natural projection of complex horizontal Laplacian on M: it is independent of the fiber coordinate. By using Sobolev space theory and spectral resolution theory in Hilbert space, we prove the Hodge theorem for the natural projection of complex horizontal Laplacian on M.     

Bochner–Weitzenböck formula and Li–Yau estimates on Finsler manifolds

We prove the Bochner–Weitzenböck formula for the (nonlinear) Laplacian on general Finsler manifolds and derive Li–Yau type gradient estimates as well as parabolic Harnack inequalities. Moreover, we deduce Bakry–Émery gradient estimates. All these estimates depend on lower bounds for the weighted flag Ricci tensor.     

Bochner-Kodaira Techniques on K¨ahler Finsler Manifolds

A horizontal Hodge Laplacian operator $\square_{\mathcal {H}}$ is defined for Hermitian holomorphic vector bundles over PTM on K¨ahler Finsler manifold, and the expression of $\square_{\mathcal {H}}$ is obtained explicitly in terms of horizontal covariant derivatives of the Chern-Finsler connection. The vanishing theorem is obtained by using the $\partial_{\mathcal {H}}\ov{\partial}_{\mathcal {H}}$-method on K¨ahler Finsler manifolds.     

强Khler-Finsler流形上(p,q)形式的中值Laplace算子(英文)

本文给出了强Khler-Finsler流形上中值Laplace算子的一些性质,如自伴性质,散度形式等。与Khler流形上利用逆变基本张量及其在Finsler流形上的变形作为密度函数定义流形上的逐点内积及整体内积不同,作者利用强Khler-Finsler流形上的逆变密切Khler度量作为密度函数定义了流形上的逐点内积和整体内积,并定义了强Khler-Finsler流形上的Hodge-Laplace算子,它可看作函数情形中值Laplace算子的推广。     

强K(a)hler-Finsler流形上(p,q)形式的中值Laplace算子

本文给出了强K(a)hler-Finsler流形上中值Laplace算子的一些性质,如自伴性质,散度形式等.与K(a)hler流形上利用逆变基本张量[11]及其在Finsler流形上的变形[5,10]作为密度函数定义流形上的逐点内积及整体内积不同,作者利用强K(a)hler-Finsler流形上的逆变密切Kahler度量作为密度函数定义了流形上的逐点内积和整体内积,并定义了强K(a)hler-Finsler流形上的Hodge-Laplace算子,它可看作函数情形中值Laplace算子的推广.