Frist Eigenvalue Of Submanifolds In Space Forms And The R-Stability Of Hypersurfaces

This paper was divided into two parts, In the first part, we use the generalized position vector field of the compact without boundary submainfold Mⁿ isometrically immersed in R^n+P(c), by establishing two integral inequalities about the tangential and the normal part of the generalized position vector of Mⁿ and the first nonzero eigenvalue λ₁^â–³ of the Laplacian operator, we give a lower bound estimate of the gap between the upper bound and λ₁^â–³. In other words, we give a better estimate of λ₁^â–³. We also give some sufficient conditions of compact without boundary submanifolds Mⁿ be immersed into a geodesic hypersphere S^n+p-1 in R^n+P(c) or isometric to a geodesic sphere Sⁿ in Kⁿ⁺¹(c), which generalized the corresponding result in Eucliclean space given by Deshmukh^[16]. In the second part, we consider the linearized operator L_r of the higher order mean curvature of a closed hypersurface immersed into a Riemannian space form Rⁿ⁺¹(c), and derive a new sharp upper bound for the first positive eigenvalue λ₁^L_r.Our bounds are extrinsic in the sense that they are given in terms of the higher order mean curvatures. We also get a lower bound of the gap between the new sharp upper bound and λ₁^L_r. Under the assumption H_r+2 > 0 and H_r+1 > 0, by establishing two valuable integral formulas, we obtain unified sharp upper bounds of λ₁^L_r respectively. We also give an estimation of the upper bounds of the first eigenvalue of Schrodinger's type operator, by which we prove those hypersurface with positive constant H_r+1 in any space forms are stable if and only if they are geodesic spheres, generalizing the previous result obtained only in the case c ≤ 0.