Eigenvalue estimates for some natural elliptic operators on hypersurfaces

Let MnMn be a compact hypersurface of a real space form and LrLr the linearized operator of the first variation of the (r+1)(r+1)th mean curvature, r∈{0,1⋯,n}r∈{0,1⋯,n}. In this paper, by a generalized Bochner-type formula for LrLr and ideas trace back to the Lichnerowicz–Obata Theorem, we obtain some sharp lower bound for the first nonzero eigenvalues of those operators LrLr.