Rigidez de hipersuperfícies que minimizam o volume em variedades riemannianas com curvatura escalar positiva

Abstract

This work extends Mendes’ Theorem [25] to higher dimensions and to the setting of Riemannian manifolds with boundary. First, we establish an inequality analogous to Mendes’ result for the volume of closed locally conformally flat hypersurfaces that minimize volume in 7-dimensional Riemannian manifolds with positive scalar curvature and nonnegative Ricci curvature. Moreover, we prove that in the equality case, the hypersurface is isometric to the 6-dimensional sphere, while the ambient manifold is isometric to a product of the form (−ε, ε) × S 6 . Additionally, we extend these results to free-boundary, totally geodesic hypersurfaces immersed in 5- or 7-dimensional manifolds with nonempty boundary and nonnegative mean curvature on the boundary. In these cases, we obtain analogous inequalities and show that, in the equality case, the hypersurface is isometric to the 4- or 6-dimensional hemisphere, while the ambient manifold is isometric to a product of the form (−ε, ε) × S 4 + or (−ε, ε) × S 6 +.