Limitante superior extrínseco para o primeiro autovalor do laplaciano

Abstract

In this work we describe a result obtained by Ernst Heintze which improves the following results due to Reilly: let M be a compact n-dimensional Riemannian manifold and λ1(M) is the first eingenvalue of laplacian. If M is isometrically immersed in λ1(M) ≤ n/vo1M ʃ M h2dM. E. Heintze, in 1988, generalized the above result. He proved that: If M is isometrically imersed in ¯M, where, K ¯M ≤ ẟ, then: (i) λ1(M) ≤ nẟ + n/vo1M ʃ M H2dM, to ẟ ≥ 0; (ii) λ1(M) ≤ nẟ + nmax H2, to ẟ < 0. Furthermore, if the equality in (i), then M is minimally immersed into some geodesic sphere.