18 apr 2016 -- 14:30
Aula di Consiglio, Dip.Matematica, Università "La Sapienza", Roma
Abstract.
I will consider the problem of prescribing the Gaussian curvature (under pointwise conformal change of the metric) on surfaces with conical singularities. This question has been first raised by Troyanov and it is a generalization of the Kazdan-Warner problem for regular surfaces, known as the Nirenberg problem on the sphere. Answer this question amounts to solve a singular differential problem on the surface. This equation has been studied first in the case K > 0, where K denotes the curvature we want to prescribe. I will present some new results (obtained in collaboration with R. Lopez-Soriano) in the case K sign-changing. When the surface is the sphere, under some mild conditions on the nodal set of K, we derived some sufficient conditions on K and on the conical singularities for the existence of solutions of (1). Even if we do not expect these conditions to be necessary, I will explain why they are somehow sharp.