Uma desigualdade integral para hipersuperfícies com curvatura escalar constante.

Abstract

In this work we will study rigidity of closed hypersurfaces (i.e., compact without border) with constant scalar curvature isometrically immersed in a Riemannian space form with constant sectional curvature. In this configuration, we will establish a Simons-type formula and, as an application, an integral inequality with the norm of the second fundamental form without a trace and a certain function depending on the scalar curvature of the hypersurface and on the sectional curvature of the ambient space. We will show that equality is achieved in this integral inequality in a totally umbilical hypersurfaces and in a certain Clifford torus, when the environment is the Euclidean sphere. In addition, we also explore this integral inequality in the case in which the ambient space is the Euclidean and the hyperbolic.