Primeiro autovalor do operador de Laplace penalizado pela curvatura média e o funcional de Willmore.

Abstract

In this work, we will prove some results for the first eigenvalue of a linear differential Schrödinger operator L = −Δ − (1/n)H*2, defined on closed hypersurfaces with the same volume of the sphere and immersed in Rn+1 , where −Δ is the Laplace-Beltrami operator and H = Pnj=1kj , with kj the hypersurface principal curvatures. Under these conditions, we will show a local generalization for the classical result of the Willmore functional for the Euclidean sphere. As a consequence, we will prove that the first eigenvalue of this operator in the Euclidean sphere is a local maximum and this result is a global one in the closed hypersurface space of R3 and genus zero.