10:00 - 11:00 - Tiarlos Cruz (Universidade Federal de Alagoas - Brazil)
On the prescribed curvature problem on the compact Riemannian manifolds with boundary
Abstract: In this talk, we will study the set of curvature functions which a given compact manifold with non-empty boundary can support. We prove that the sign demanded by the Gauss-Bonnet Theorem is a necessary and sufficient condition for a given function to be the geodesic curvature or the Gaussian curvature of some conformally equivalent metric. This is based on a joint work with A. Santos and F. Vitório.
11:30 - 12:30 - Vanderson Lima (Universidade Federal do Rio Grande do Sul - Brazil)
Eigenvalue problems and free-boundary minimal surfaces in spherical caps
Abstract: In a recent work with Ana Menezes (Princeton University), we introduced a family of functionals on the space of Riemannian metrics of a compact surface with boundary, defined via eigenvalues of a Steklov-type problem. In this talk I will present the ideas behind the following results: each such functional is uniformly bounded from above, and the maximizing metrics are induced by free boundary minimal immersions in some geodesic ball of a round sphere; the maximizer in the case of a disk is a spherical cap of dimension two; free boundary minimal annuli in geodesic balls of round spheres which are immersed by first eigenfunctions are rotationally symmetric.
14:15 - 15:15 - Leandro F. Pessoa (Universidade Federal do Piauí - Brazil)
Infinity harmonic functions and the splitting of complete manifolds
Abstract: We will start the talk given a brief motivation for the study of splitting results under functional assumptions. Our main goal is to prove a splitting result for a surface whose curvature is non-negative and that supports a non-constant infinity harmonic function with at most linear growth on one side. Moreover, in this case the infinity harmonic function is shown to be linear. This is a joint work with D. Araujo, M. Magliaro and L. Mari.