Maximizando o primeiro autovalor do operador de Jacobi

Abstract

We consider the Jacobi operator, defined on a closed oriented hypersurfaces immersed in the Euclidean space with the same volume of the unit sphere by L = −∆−|II|2, where −∆ is the Laplace-Beltrami operator with ∆u = div(∇u) and |II| 2 = ∑nj = 1k2j is the square of second fundamental form. We show a generalization for the classical result of the Willmore functional for the Euclidean sphere. As a consequence, by adding a topological hypothesis we prove that the fi rst eigenvalue of the Jacobi operator in the Euclidean sphere is a global maximum.