Sobre as condições de Ricci para imersões de curvatura média constante de fronteira livre em bolas de formas espaciais

Abstract

In this dissertation, we investigate the Ricci conditions for immersions of constant mean curvature and free boundary on balls of space forms. Given a Riemannian manifold Σ 2 with a metric ds2 whose Gaussian curvature is Ks < H2 0 +c, the necessary condition and sufficient of Gregorio Ricci-Curbastro so that Σ 2 is isometrically immersed as a surface of constant or minimum mean curvature in balls of space form and that new metric ds˜ 2 = −Ks + H2 0 + c ds2 is flat. In the same meaning, we have that the existence of minimum immersions in R 3 , the Simons Type equation, for the three-dimensional case, is equivalent to the differential equation Ks △ Ks − ||∇Ks||2 − 4K 3 s = 0 with Ks < 0 and is generalized for immersions of constant mean curvature in space forms by the equation: (−Ks + H 2 0 + c0) △ Ks + ||∇Ks||2 + 4(−Ks + H 2 0 + c0) 2Ks = 0. Adding such a condition to a minimum isometric immersion f : Σ 2 → B n, with possible branch points and without umbility points, we show that after a possible reduction of codimension, f(Σ 2 ) is essential in R 3 or essential in R 6 . We thus obtain an analytical version for the minimum immersion f, with possible branch points, where f(Σ 2 ) meets ∂B orthogonally, so that f(Σ 2 ) is totally umbilical.