Upper bound for the first eigenvalue of algebraic submanifolds

Abstract

1. Statement of results Let M be a compact manifold endowed with a Riemannian metric. The spectrum of the Laplacian, A, acting on functions form a discrete set of the form {0 < ),~ < 22 < �9 �9 �9 < 2k < �9 �9 �9 -~ ~}. In 1970, Joseph Hersch [5] gave a sharp upper bound for the first non-zero eigenvalue 2~ for any Riemannian metric on the 2-sphere in terms of its volume alone. Similar estimates for 21 on any compact oriented surfaces were derived by Yang and Yau [7]. The second and the third authors [6] studied the non-orientable surfaces and pointed out the relationship of 21 and the conformal class of the surface. In fact, their estimates were applied to study the Willmore problem. Another application of these types of upper bounds was found by Choi and Schoen [3] in relation to the set of all minimal surfaces in a compact 3-manifold of positive Ricci curvature. The purpose of this paper is to prove a higher dimensional generalization of the above results. It was pointed out by Marcel Berger [l] that Hersch's theorem fails in higher dimensional spheres. In view of the relationship between 21 and the conformal structure of a surface as indicated by Li and Yau [6], we were thus motivated to study the complex category.