Sobre teorema de comparação de autovalores de Cheng

Abstract

We present a version of Cheng’s Eigenvalue Comparison Theorem, where the limitation of the sectional and Ricci curvature is changed by limiting the mean curvature of the ball away. Furthermore, the present construction of smooth metrics gk,in [0,r] x S3, non-isometric to the canonical metric of constant sectional curvature k, cank,such that the balls geodesic Bgk(r)=([0,r]x S3,gk),Bcank(r)=([0,r]x S3,cank) have the same first eigenvalue, the same volume and the distances spheres ӘBgk(s)and ӘBcank(s),0 < s ≤ r, has the same mean curvature. Finally, this version of Cheng’s Eigenvalue Comparison Theorem to construct examples of Riemannian manifolds M with positive fundamental tone.