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.

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.