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几何 >> 黎曼几何 >> 非正曲率和负曲率流形
Questions in category: 非正曲率和负曲率流形 (Nonpositively and negatively curved manifolds).

1

Thm[Beckenbach and Rado]

Posted by haifeng on 2014-07-29 10:26:20 last update 2014-07-29 10:29:45 | Answers (0) | 收藏

Thm[Beckenbach and Rado] 设 $(M,g)$ 是一完备黎曼曲面, Gauss 曲率非正, 则

\[
|\partial\Omega|_g^2\geqslant 4\pi|\Omega|_g,
\]

并且仅当 $(M,g)$ 是欧氏平面, $\Omega$ 是一圆盘时等号成立.

 

References:

[1] E. F. Beckenbach and T. Rado, Subharmonic functions and surfaces of negative curvature. Trans. Amer. Math. Soc. 35 (1933), 662–674.

2

[Conjecture]任意闭的可见黎曼流形都存在一个度量,使得截面曲率小于等于-1.

Posted by haifeng on 2013-03-03 21:14:00 last update 2013-03-03 21:16:20 | Answers (0) | 收藏

Conjecture: Any closed visibility manifold $M^n$ must admit a metric $g$ of negative sectional curvature $\text{sec}_g\leq −1$.


Reference:

Jianguo Cao and Xiaoyang Chen, Minimal volume and simplicial norm of visibility n-manifolds and compact 3-manifolds. arXiv:0812.3353v4.


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