A proof of the Alexanderov's uniqueness theorem for convex surfaces in $\mathbb R^3$

TL;DR

This paper presents a new proof of Alexandrov's classical uniqueness theorem for convex surfaces in three-dimensional space, utilizing the Bers-Nirenberg weak uniqueness continuation theorem under minimal regularity assumptions.

Contribution

It introduces a novel proof method for Alexandrov's theorem based on PDE techniques, reducing regularity requirements for convex surface uniqueness.

Findings

Established a new proof of Alexandrov's theorem using PDE methods.

Reduced regularity assumptions needed for the theorem.

Extended the theorem to convex bodies with Radon measure spherical Hessians.

Abstract

We give a new proof of a classical uniqueness theorem of Alexandrov using the weak uniqueness continuation theorem of Bers-Nirenberg. We prove a version of this theorem with the minimal regularity assumption: the spherical hessians of the corresponding convex bodies as Radon measures are nonsingular.