## Uniqueness of immersed spheres in three-manifolds. Proof of a conjecture by Alexandrov

Type:
Site:
Date:
11/04/2016 - 15:30 - 16:30
Salle:
2007
Orateur:
MIRA Pablo
Localisation:
Université polytechnique de Carthagène
Localisation:
Espagne
Résumé:

A famous theorem by Hopf proves that any constant mean curvature sphere in $\mathbb R^3$ is a round sphere. In this talk we will generalize Hopf's theorem to classes of surfaces modeled by arbitrary elliptic PDEs in arbitrary three-manifolds, with the only hypothesis of the existence of a family of "candidate surfaces" within the class. In this way, we prove that any immersed sphere in such a class of surfaces is a candidate sphere.

As an application, we prove a 1956 conjecture by A.D. Alexandrov on the uniqueness of immersed spheres of prescribed curvatures in $\mathbb R^3$, and we complete the characterization of round spheres as the only elliptic Weingarten spheres in $\mathbb R^3$ (Weingarten spheres are immersed spheres in $\mathbb R^3$ whose principal curvatures are linked by a non-trivial elliptic relation).

This is a joint work with J.A. Gálvez.