Geometry & Topology seminar

Xiangwen Zhang, Columbia University 

“ALEXANDROV’S UNIQUENESS THEOREM FOR CONVEX SURFACES”

A classical uniqueness problem of Alexandrov says that a closed, strictly convex twice differentiable surface in R3 is uniquely determined to within a parallel translation when one gives a proper function of the principal curvatures. We will talk about a PDE proof of this theorem, by using the maximum principle and a weak uniqueness continuation theorem of Bers- Nirenberg. Moreover, a stability result related to the uniqueness problem will be mentioned. This is joint work with P. Guan and Z. Wang. If time permits, we will also briefly introduce the idea of our recent work on Alexandrov’s theorems for codimension two submanifolds in spacetimes.