Description:

Title: Symmetry of hypersurfaces and the Hopf Lemma

In 1945, S.S. Chern provided the following characterization of spheres in three-dimensional Euclidean space:  
Let $M$ be a closed convex surface satisfying
$F(\kappa_1, \kappa_2) = 1$ where $\kappa_1$ and $\kappa_2$ denote the principal curvatures, and $F$ is elliptic in the sense that $\partial_{\kappa_i} F > 0$. Then $M$ must be a sphere.  

Important special cases include $F(\kappa_1, \kappa_2) = \kappa_1 + \kappa_2$ and $F(\kappa_1, \kappa_2) = \kappa_1 \kappa_2,$ corresponding to prescribed mean curvature and prescribed Gaussian curvature, respectively.

Nirenberg and I explored extensions of this problem and proposed the following conjecture:  
Let $M$ be a closed convex surface in three-dimensional Euclidean space, and let $F$ be elliptic. Suppose that for any two points $(X_1, X_2, X_3)$ and $(X_1, X_2, \hat X_3)$ on $M$ with $X_3 \ge \hat X_3$, the inequality $F(\kappa_1, \kappa_2)(X_1, X_2, X_3) \le F(\kappa_1, \kappa_2)(X_1, X_2, \hat X_3)$ holds. Then $M$ must be symmetric about some hyperplane $X_3=constant$.

In this talk, I will survey developments in this area and present open problems, both related to resolving this conjecture and to broader conjectures concerning extensions of the Hopf Lemma.