A sharp upper bound for the first eigenvalue of the Laplacian of compact hypersurfaces in rank-1 symmetric spaces

摘要

Let M be a closed hypersurface in a simply connected rank-1 symmetric space M. In this paper, we give an upper bound for the first eigenvalue of the Laplacian of M in terms of the Ricci curvature of M and the square of the length of the second fundamental form of the geodesic spheres with center at the center-of-mass of M.