Estimativas do primeiro autovalor do laplaciano e caracterização de hipersuperfícies isoparamétricas em Sn+1 - Barbosa, José Nelson Bastos (2002)
This work consists of three chapters addressing different subjects about compact hypersurfaces of the unit Euclidean sphere Sn + 1. In the first chapter, we will prove that a compact hypersurface of Sn + 1 with non-negative Ricci curvature and infinite fundamental group is isometric to a torus of constant mean curvature. Next, we will consider the surface hypersurfaces of Sn + l with two different main curvatures and also approach a case in which one main curvature has the signal different from the others. In the second chapter, we will apply the Minakshisundaram-Pleijel heat core expansion formula to characterize compact hypersurfaces of Sn + 1 through the Laplacian spectra of the p-forms, for some p values. Finally, in the third chapter, we will use the Bochner-Lichnerowicz formula to obtain estimates of the norm of the second fundamental form of a compact minimal hypersurface of Sn + 1 as a function of first eigenvalue of the Laplacian and the dimension of Mn, and, using formula of Reilly, we obtain an estimate for the first eigenvalue of the Laplacian of a closed and dipped hypersurface of a compact Riemannian manifold whose Ricci curvature is bounded lower by a positive constant.