问题

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

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.