Estimativas para o índice de superfícies compactas em 3-variedades com a propriedade de Killing

Abstract

In this thesis we study two distinct topics. First, we obtain Morse index estimates for minimal or constant mean curvature compact surfaces M2 immersed in 3-manifolds M ̄ 3 satisfying the Killing property, that is, manifolds that admit an orthonormal frame of Killing vector fields. Specifically, we prove that Morse index of these surfaces is bounded from below by a linear function of its topological genus. In the second topic we consider a compact free-boundary submanifold M ̄ n of a Riemannian manifold M ̄ n+k , n ≥ 3, positively curved and with convex boundary. We prove that there exists an explicit constant C(n), such that if the norm of the traceless second fundamental form Φ of M is bounded by C(n) then the first cohomology group H1 (M) vanishes. In particular, M has only one boundary component. Moreover, if k = 1, under suitable conditions on the curvature of M ̄ n+1 and on the convexity of its boundary we prove that for 2 ≤ p ≤ n − 2, there exists a constant C(n, p) such that if |Φ| ≤ C(n, p) then Hp (M) vanishes.