Banca de DEFESA: ANTONIO DEÍGERSON DA COSTA LOPES

Uma banca de DEFESA de MESTRADO foi cadastrada pelo programa.
STUDENT : ANTONIO DEÍGERSON DA COSTA LOPES
DATE: 26/08/2022
TIME: 09:30
LOCAL: Videoconferencia
TITLE:

On Ricci's Conditions for Immersions of
Constant Mean Curvature of Free Boundary in
Balls of Space Shapes.

KEY WORDS:

Ricci conditions, free boundary immersions, surfaces of constant mean curvature

PAGES: 65
BIG AREA: Ciências Exatas e da Terra
AREA: Matemática
SUBÁREA: Geometria e Topologia
SPECIALTY: Geometria Diferencial
SUMMARY:

In this dissertation, we will investigate the Ricci conditions for immersions of constant mean curvature of free boundary in balls of space forms. Given a two-dimensional Riemannian manifold $\Sigma^{2}$ with a metric $ds^{2}$ whose Gaussian curvature is $K_{s}< H^{2}+c$, the Gregory Ricci-Curbastro condition is a necessary and sufficient condition for such an immersion to be isometrically minimally immersed or of constant mean curvature in a space form is that the new metric $d\Tilde{s}^{2}=\sqrt{-K_{s}+H^ {2}+c}ds^{2}$ be flat. If the condition for performing this immersion is that $\Sigma^{2}$ is simply connected, then we can perform the immersion induced by the metric $ds^{2}$ on balls of space forms. In the same vein, we saw that the existence of minimal immersions in $\mathbb{R}^{3}$, the Simons Type equation, for the three-dimensional case, is equivalent to the differential equation $K\bigtriangleup K - ||\nabla K ||^{2} - 4K^{3}=0$ with $K<0$. Adding such a condition to a minimal isometric immersion $f: {\Sigma}^{2} \rightarrow B^{n} $, with possible branch points and no umbility points, we show that after a possible codimension reduction, $f (\Sigma^{2})$ essentially arrives at $\mathbb{R}^{3}$ or essentially $\mathbb{R}^{6}$. We thus obtain an analytic version for the minimal immersion $f$ , with possible branch points, where $f(\Sigma^{2})$ meets $\partial B $ orthogonally, so that $f(\Sigma^{ 2})$ is totally umbilical.

BANKING MEMBERS:
Presidente - 1165266 - FELICIANO MARCILIO AGUIAR VITORIO
Interno - 2474631 - MARCIO HENRIQUE BATISTA DA SILVA
Externo à Instituição - NEWTON LUIS SANTOS - UFPI