A new characterization of submanifolds with parallel mean curvature vector in $S^{n+p}$

Abstract

In this work we will consider compact submanifold M n immersed in the Euclidean sphere S nþp with parallel mean curvature vector and we introduce a Schrö dinger operator L ¼ ÀD þ V , where D stands for the Laplacian whereas V is some potential on M n which depends on n; p and h that are respectively, the dimension, codimension and mean curvature vector of M n . We will present a gap estimate for the first eigenvalue m 1 of L, by showing that eitherAs a consequence we obtain new characterizations of spheres, Cli¤ord tori and Veronese surfaces that extend a work due to Wu [W] for minimal submanifolds.

Spectral characterization and Schrödinger operator of space-like submanifolds

“…Since matrices Φ α and Φ n+1 are traceless, by Lemma 2.1, we have 6) where the following 7) is used. By the Cauchy-Schwarz inequality, we have…”

“…We state a Proposition which can be proved by making use of the similar method due to C. Wu [17] or A.A. Barros et al [6] for Riemannian manifold. (1).…”

“…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. In connection with the stability for hypersurfaces with constant mean curvature or constant scalar curvature in Riemannian manifolds, Barbosa-Oliker [4] and Liu-Deng [10] studied the stability for space-like hypersurfaces with constant mean curvature or constant scalar curvature in Lorentz manifolds.…”

First eigenvalue of Schrodinger operator of space-like hypersurfaces

“…We firstly state a proposition proved by making use of a method similar to that used by C. Wu [16] or A. A. Barros et al [8] for a Riemannian manifold.…”