We establish integral formulas and sharp two-sided bounds for the Ricci curvature, mean curvature and second fundamental form on a Riemannian manifold with boundary. As applications, the Lichnerowicz estimate and Obata theorem for the first eigenvalue, as well as sharp gradient/Hessian estimates on the boundary for eigenfunctions, are presented for the Dirichlet and Neumann eigenproblems. 查看全文>>