Sharp upper bounds for $\lambda _1^{L_r }$ of immersed hypersurfaces and their stability in space forms

Abstract

In this paper we derive new sharp upper bounds for the first positive eigenvalue $\lambda _1^{L_r }$ of the linearized operator of the higher order mean curvature of a closed hypersurface immersed into a Riemannian space form ℝⁿ⁺¹(c) (c > 0). Our bounds are extrinsic in the sense that they are given in terms of the higher order mean curvatures. Under the assumption H _r+2 > 0, by establishing two valuable integral formulas, we obtain unified sharp upper bounds of $\lambda _1^{L_r }$. We also give an estimation of the upper bounds of the first eigenvalue of a Schrödinger-type operator, by which we prove those hypersurfaces with positive constant H _r+1 in any space forms are stable if and only if they are geodesic spheres, thus generalizing the previous result obtained only in the case c ≤ 0.