Estimando o primeiro autovalor do Laplaciano de superfícies

Abstract

In the article On the first Laplacian eigenvalue for compact submanifolds of Euclidean space [11], Reilly presents three geometric estimates for the first Laplacian eigenvalue of a compact embedded hypersurface of the Euclidean space. In this work, using basic concepts of Differential Geometry and Geometric Analysis we will prove the same results in the setting of surfaces of R 3. We follow the same strategy to characterize spheres as the only connected compact surfaces whose its Gaussian curvature is bounded from above by the first Laplacian eigenvalue, such result is due to Deshmukh in the article Compact Hypersurfaces in a Euclidean Space [5].