Rigidez de esferas mínimas com área 4π

Abstract

In this work, we will prove two theorems due to Mazet and Rosenberg in (Mazet; Rosenberg, 2014). These results characterize a complete Riemannian 3-manifold M under certain conditions. The first theorem requires that a sectional curvature of M satisfies 0 ≤ K ≤ 1, and states that if a minimal embedded 2-sphere Σ in M has area |Σ| equal to 4π, then M is isometric to a canonical sphere (S 3 , gcan) with sectional curvature equal to 1 or a quotient of the product S 2 × R. The second theorem is a rigidity theorem for hyperbolic cusps in which M has sectional curvature K ≤ −1, and states that if T is a torus of constant mean curvature equal to 1 embedded in M then the convex side of T in M is isometric to T 2 × R+(hyperbolic cusp).