Rigidez, não estabilidade e estimativa inferior para o índice de Morse de subvariedades mínimas e CMCs em variedades conformemente Euclidianas

Abstract

This work has three main goals. The first consists of investigating some gap, index and non-stability results for f-minimal submanifolds. The second one is to investigate stability, as well as non-stability criteria, for hypersurfaces of constant mean curvature (CMC) that intersect at a constant angle a Euclidean ball provided with a certain conformal metric. The second is to study classification and Morse Index of minimal hypersurfaces in space forms. In the first part of this thesis we will study, in the context of weighted Riemannian manifolds, some gap results, we will also look for a lower bound for the Morse index and non-stability theorems for closed and free-bound f-minimal submanifolds. In the second part, we obtained two non-stability criteria. The first criterion, for capillary hypersurfaces, uses the region surrounded by the hypersurface and the “wetting area” of the ball, denoted as a generalized region, we obtain that if such region is symmetrical with respect to the origin then the hypersurface is not stable. The second criterion states that if the normal field of a free boundary CMC hypersurface has zero mean then such hypersurface is not stable. In the third part, we determine that the Morse index of minimal free boundary surfaces in hyperbolic and spherical space is at least 4. Furthermore, still in hyperbolic and spherical space, we conclude that if a surface reaches Morse Index 4, then the first Steklov’s eigenvalue is equal to the principal curvature of the hyperperbolic and spherical ball respectively. For the space forms, we also obtained that, among capillary minimal hypersurfaces, the totally geodesic ones are characterized by a condition of pinching in the length of their second fundamental form.