Rigidez de superfícies minimizantes de área em variedades tridimensionais

Abstract

In this work, we present a demonstration from (MICALLEF; MORARU, 2015), where the authors unify the proofs of results previously established by Cai and Galloway (CAI; GALLOWAY, 2000), Bray-Brendle-Neves (BRAY; BRENDLE; NEVES, 2010), and Nunes (NUNES, 2013). In these articles, lower bounds are imposed on the scalar curvature of the ambient space, which is a three-dimensional Riemannian manifold, to derive geometric and topological conclusions about the minimal surfaces contained within it. More precisely, we define R0 := minx∈MR(x), where R0 ∈ {−2, 0, 2}, and assume certain conditions regarding the area and, in two of the cases, the genus of the surface Σ. Thus, we draw conclusions about the metric of the ambient manifold in a neighborhood of Σ.