Sobre hipersuperfícies mínimas, aplicações do princípio do máximo fraco e de teoremas tipo-Liouville

Abstract

In this work we approach four research lines, where we began with the study of isometrically immersed hypersurfaces in a horoball. Next we studied Liouville type theorems in a complete Riemannian manifold for general operators. After we studied hypersurfaces f-minimal closed on a manifold with density, and nally we studied properly embedded minimal hypersurfaces with free boundary in a n-dimensional compact Riemannian manifold. Continuing, we obtain under a more general class operator than '-Laplacian, a Liouville type theorem for a complete Riemannian manifold, so that, prove a classi cation theorem for Killing graph of a foliation. Firstly, we are going to assume a weak maximum principle and that immersion is contained in a horoball, i.e., the set of bounded above Bussemann functions . We obtain an estimate for the highest quotient of r-curvatures. Moreover, under certain conditions on sectional curvature and assuming that the immersion is contained in a horoball, we forced the validity of the weak maximum principle and obtain the same estimates. Next, we establish a Choi-Wang type estimate for the rst eigenvalue of the weighter Laplacian on spaces with density in responding partially to Yau's conjecture for the rst eigenvalue weighter Laplacian for spaces with density, and moreover, we obtain an inequality Poincar e type. With the estimates obtained, we establish an estimate of volume for a closed surface immersed in a space with density. Still following the study of spaces with density, we obtain a type Hientze-Karcher inequality for a compact manifold with nonempty boundary , so that, we obtain that if holds the equality than the manifold is isometric to a Euclidian ball. As consequence, we obtain under same conditions that if the f-mean curvature satisfy a bounded below than the manifold is isometric to a Euclidian ball. Finally, we obtain an estimate for the nonzero rst Steklov eigenvalue, where we are giving a answer partial to a conjecture by Fraser and Li. Moreover, as a consequence we establish an estimate for the total length of the boundary of the properly embedded minimal surfaces with free boundary in terms of its topology, thus, we proved the same when the surface is embedded in the Euclidean ball 3-dimensional.