Curvatura gaussiana prescrita em superfícies com característica de Euler negativa

Abstract

This dissertation aims to study the problem of prescribing the Gaussian curvature on closed Riemannian surfaces with negative Euler characteristic, detailing the results obtained in [3]. The work investigates the existence of multiple solutions to the problem of finding conformal Riemannian metrics whose Gaussian curvature equals a given function. Under certain conditions, the problem admits a unique solution, which corresponds to the global minimum of a specific functional. By considering small perturbations to a parameter of the function intended to be prescribed as the Gaussian curvature, it is shown that the functional admits an additional critical point of the “mountain pass” type. Furthermore, the behavior of this second solution is investigated as the parameter approaches zero.