A proof of Alexandrov's uniqueness theorem for convex surfaces in \( R^{3} \)

Abstract

We give a new proof of a classical uniqueness theorem of Alexandrov [4] using the weak uniqueness continuation theorem of Bers-Nirenberg [8]. 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.

The Weyl problem in warped product spaces

“…In the Euclidean space, we give two proofs only using maximum principle. The idea comes from [15] and [18]. For hyperbolic and spherical space cases, we use Beltrami map to extend the infinitesimal rigidity in Euclidean space to space forms.…”

The Rigidity of Hypersurfaces in Euclidean Space

“…By the Lemma 4 in [7], we have, (a 1 1 ) 2 + (a 1 2 ) 2 + (a 2 1 ) 2 + (a 2 2 ) 2 ≤ −C det(a j i ). We conclude that,…”

Linearity of homogeneous solutions to degenerate elliptic equations in dimension three

“…This solved an open problem by Safonov [19]. Specifically, if 0 < λ(x) ≤ Λ(x) are the smallest and largest eigenvalues of (a ij (x)), and we denote K(x) := Λ(x)/λ(x) ≥ 1, Han, Nadirashvili and Yuan imposed the condition An alternative proof of Theorem 1.1 was obtained in 2016 by Guan, Wang and Zhang [5], again under very weak regularity assumptions on u. For that, they treated the problem directly as a uniformly elliptic equation in S 2 , and gave an elegant argument using the Bers-Nirenberg unique continuation theorem.…”