Hipersuperfícies mínimas de fronteira livre em alguns domínios euclidianos e outros tópicos

Abstract

This thesis has two main goals. The first of them is to inquire the existence of free boundary minimal hypersurfaces in quadric domains of R n. The second objective is to study the Weingarten hypersurfaces in R n with two distinct principal curvatures, one of them being simple. In the first part of the thesis, we have reached results concern existence, non- existence and unicity of free boundary minimal hypersurfaces in several quadric domains. One of these results characterizes the totally geodesic disks as the only free boundary compact minimal hypersurfaces in unbounded domains. We have obtained a classification result for free boundary minimal surfaces in the unit ball of R 3 . We have imposed that a linear dependency that is always verified on the surface’s boundary would be extended to a collar over a boundary component. In the end of this part, we have used some geometric identities obtained so far to reach a new gap result for free boundary minimal hypersurfaces in the unit ball of R n and to study the existence of free boundary minimal hypersurfaces in rotational graphs of R n . The second part of the thesis deals with Weingarten hypersurfaces Mn−1 ⊂ R n in the way that a linear combination of two higher order mean curvatures is constant, i.e., aHr + bHs = 1, a, b ∈ R with a 2 + b 2 6= 0. We have studied Weingarten hypersurfaces with two distinct principal curvatures, one of them with multiplicity 1. In this context, we have obtained two results assuring isometry among M and the cylinder R × S n−2 (ρ) under certain conditions involving a, b, r, s and some additional restriction on a k-mean curvature.