Rigidity of compact manifolds with boundary and nonnegative Ricci curvature

Let $\overline {\Omega }$ be an ($n+1$)-dimensional compact Riemannian manifold with nonnegative Ricci curvature and nonempty boundary $M=\partial \overline {\Omega }$. Assume that the principal curvatures of $M$ are bounded from below by a positive constant $c$. In this paper, we prove that the first nonzero eigenvalue $\lambda _{1}$ of the Laplacian of $M$ acting on functions on $M$ satisfies $\lambda _{1} \geq nc^{2}$ with equality holding if and only if $\Omega$ is isometric to an $(n+1)$-dimensional Euclidean ball of radius $\frac {1}{c}$. Some related rigidity theorems for $\overline {\Omega }$ are also proved.