A proof of the Alexanderov's Uniqueness Theorem for Convex Surfaces in R3

A proof of the Alexanderov's Uniqueness Theorem for Convex Surfaces in R3

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Xiangwen Zhang , Columbia University
Fine Hall 314

PLEASE NOTE SPECIAL DAY (WEDNESDAY).  CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.  A classical uniqueness theorem of Alexandrov says that: if M and M_0 are two closed strictly convex C^2 surface in R^3 and satisfy f(a,b) = f(c,d), at points of M, M_0 with parallel normals, for some C^1 function f(x,y) with \partial_{x}f\partial_{y}f>0, then M is equal to M_0 up to a translation. We will talk about a new PDE proof for this theorem by using the maximal principle and weak uniqueness continuation theorem of Bers-Nirenberg. More generally, we prove a version of this theorem with the minimal regularity assumption: the spherical hessians of the supporting functions for the corresponding convex bodies as Radon measures are nonsingular. This is a joint work with P. Guan and Z. Wang.