Hipersuperfícies com bordo livre e rigidez de superfícies mínimas

Abstract

In this thesis, we prove estimates for the volume and boundary area of stable hypersurfaces ∑n-1 with nonpositive Yamabe invariant satisfying the free boundary condition in a Riemannian manifold Mn with bounds for the scalar curvature and the mean curvature of the boundary. Assuming further that ∑ is locally volume-minimizing in a manifold M with scalar curvature bounded below by a nonpositive constant, we conclude that locally M splits along ∑ as (-Є, Є)x ∑, for some Є > 0. In the case that ∑ locally minimizes a certain functional inspired by the work of Yau (2001), a neighborhood of ∑ in M is isometric to ((-Є, Є) x ∑, dt2 + e2tg), where g is Ricci at. In the second part, we study other scalar curvature rigidity phenomena adapting a technique developed by Máximo e Nunes (2013) to show a local rigidity result for three-dimensional Riemannian manifold M3 whose scalar curvature is bounded from below by a negative constant. We prove the following result: Let ∑2 ⊂ M3 be a stable minimal surface which locally maximizes the Hawking mass on M. Then M near ∑ is a piece of one the Kottler space.