Prescrição de curvaturas gaussiana e geodésica em superfícies compactas com característica de Euler não positiva

Abstract

This dissertation, based on the article [30], addresses the problem of prescribing Gaussian and geodesic curvatures on a compact Riemannian surface with boundary (Σ, g). The main goal of this work is to prove, in broad terms, the existence of a conformal metric g = e u g such that the Gaussian and geodesic curvatures with respect to g are prescribed. The problem is reduced to finding a solution for a second-order elliptic partial differential equation with boundary conditions. Furthermore, the work explores the energy functional associated with this problem and its variational properties, addressing issues of coercivity and the existence of minimizers. The study focuses on different scenarios, including cases where the Euler characteristic of Σ is negative or zero. Additionally, there exists a function defined on ∂Σ, denoted by D, which plays a fundamental role in the study. When D(q) > 1 for some point q E ∂Σ, the analysis becomes more delicate. The work employs concepts from Riemannian Geometry, Functional Analysis, and Partial Differential Equations to develop the proofs of the main theorems. Moreover, advanced methods such as blow-up analysis and the Morse index of solutions are used to handle more refined situations.