L. Nirenberg
Courant Institute

A Geometric Problem and the Hopf Lemma

Abstract: Let $ U$ be a bounded convex body in $ \mathbb{R}_n$ with smooth boundary $ M$. Assume that for any two points $ (x',a)$ and $ (x',b)$ on $ M$, with $ a<b$ the mean curvature of $ M$ at the first is not less than that at the second. Under some additional condition we show that $ M$ is symmetric about a hyperplane $ x_n$ = constant. Relations to generalized Hopf Lemma are discussed.