Contribuições à geometria espectral de superfícies: estimativas de autovalores e teoremas de rigidez

Abstract

In this thesis, we obtain upper bounds for the first eigenvalue of the stability operator of constant mean curvature (CMC) hypersurfaces with free boundary. As an application, we obtain rigidity results for the area of CMC hypersurfaces under conditions on the first eigenvalue and on the curvature of the ambient space. By changing the boundary condition, we obtain an estimate for the first eigenvalue of the Jacobi-Steklov problem and, as an application, we obtain a rigidity result involving the length of the hypersurface boundary. We also comment on some of these results in the weighted context. We note that part of the results still applies to the principal eigenvalue of a MOTS immersed in a 3-manifold M of spacelike in a spacetime if the dominant energy condition is imposed and a condition on the curvature of M is met. We establish the second variation of the area for spacelike hypersurfaces with free boundary immersed in a spacetime. Furthermore, we obtain conditions for this immersion to be stable in a Robertson-Walker spacetime. Finally, we present two rigidity results for stable CMC surfaces immersed in 3-manifolds with positive Ricci curvature when the Hawking mass is zero.