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$\olm$. In this paper, we give an upper bound for the first eigenvalue of theLaplacian of $M$ in terms of the Ricci curvature of $\olm$ and the square ofthe length of the second fundamental form of the geodesic spheres with centerat the center-of-mass of $M$.