A new characterization of submanifolds with parallel mean curvature vector in Sn+p

Kodai Mathematical Journal

Abdênago Alves de Barros¹  Aldir Chaves Brasil Jr.¹  Luis Amancio Machado de Soursa Jr.¹ 

关键词: Mean curvature vector;    first eigenvalue;    Clifford torus;   

DOI  :  10.2996/kmj/1085143788

学科分类：数学（综合）

来源: Tokyo Institute of Technology, Department of Mathematics

【 摘 要 】

References(8)In this work we will consider compact submanifold Mn immersed in the Euclidean sphere Sn+p with parallel mean curvature vector and we introduce a Schrödinger operator L=−Δ+V, where Δ stands for the Laplacian whereas V is some potential on Mn which depends on n, p and h that are respectively, the dimension, codimension and mean curvature vector of Mn. We will present a gap estimate for the first eigenvalue μ1 of L, by showing that either μ1=0 or μ1≤−n(1+H2). As a consequence we obtain new characterizations of spheres, Clifford tori and Veronese surfaces that extend a work due to Wu [W] for minimal submanifolds.

【 授权许可】

Unknown