关于R^(n)中的几何容度的刚性定理

On rigidity theorems by geometric capacity

本文研究R^(n)中开子集的由几何容度刻画的几何性质,并证明两个刚性定理:若区域在其某一边界点处的任一尺度之下都具有相应的锥的容度,则该区域必定是具有相同立体角的锥体;若区域的任意边界点均满足如上容度条件,则该区域只能为半空间.

Local capacity is introduced to prove that a subset in R^(n) with the same local capacity around a boundary point as that of a cone must be a cone of the same solid angle.Consequently,we prove the rigidity of half spaces using characterizations by the local capacity.