Stable hypersurfaces with constant scalar curvature

Abstract

Abstract.We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegative constant scalar curvature in Euclidean spaces. As a special case we prove that there is no complete noncompact strongly stable hypersurface M in R 4 with zero scalar curvature S 2 , nonzero GaussKronecker curvature and finite total curvature (i.e. R M |A| 3 < +∞).

Cones in the Euclidean space with vanishing scalar curvature

“…We will also need the following lemma whose proof uses Lemmas (3.7) and (4.1) in (Alencar et al 1993 Proof of the Lemma: From Lemmas 2 and 3 we have…”

Spectral characterization and Schrödinger operator of space-like submanifolds

“…In analogy with the case of constant mean curvature, questions of stability can be considered for hypersurfaces with constant scalar curvature. In [2], Alencar, do Carmo and Colares extended the study of stability to hypersurfaces with constant scalar curvature. As researched in C. Wu [17] for minimal submanifolds in a unit sphere, A.A. Barros et al [6] and Cheng [8] studied the first eigenvalues of some Schrödinger operators of submanifolds with parallel mean curvature vector or hypersurfaces with constant scalar curvature in a unit sphere and obtained some spectral characterizations of so called Veronese surface, Clifford torus or Riemannian product S m (r) × S n−m ( √ 1 − r 2 ), 1 ≤ m ≤ n − 1.…”

First eigenvalue of a Jacobi operator of hypersurfaces with a constant scalar curvature

“…where ∆ stands for the Laplace-Beltrami operator, arose naturally in the study of the stability of both minimal hypersurfaces in S n+1 (1) and hypersurfaces with constant mean curvature in S n+1 (1). The J m is called a Jacobi operator or a stability operator, which represents the second variation of the volume.…”

“…The differential operator was introduced and used by S. Y. Cheng and Yau in [11] to study compact hypersurfaces with constant scalar curvature in S n+1 (1). They proved that if M is an n-dimensional compact hypersurface with constant scalar curvature n(n − 1)r, r ≥ 1, and if the sectional curvature of M is non-negative, then M is a totally umbilical hypersurface S n (c) or a Rie-…”