The prescribed Gaussian curvature problem

The prescribed Gaussian curvature problem
Lugar
Aula Profesor Antonio de Castro Brzezicki, Edificio Celestino Mutis
Autor
David Ruiz Aguilar
Tipo de evento
Descripción

Coloquio "José Mendoza Ríos" IMUS-IEMath-GR.

Given a smooth function on a compact surface, does there exist a conformal metric realizing that function as its curvature? This is a classical problem in Geometric Analysis that goes back to the works of Berger, Kazdan & Warner and Moser in the 70's.

This question leads to a semilinear elliptic problem with a exponential nonlinearity, which has a variational structure. This kind of problems appear also in very different contexts: for instance, in the abelian Chern-Simons theory or in the Electroweak theory. In this talk I will try to give an overview on the topic, that still yields intricate open problems. In particular, the case of the sphere (the so-called Nirenberg problem) is not yet completely understood.

After that we will adress the same question on a surface with conical points, or with boundaries. We will show that, in some cases, the corresponding Euler-Lagrange functional is not bounded from below . Therefore we shall look for saddle-points of the functional, with the help of min-max theory.