## Kodai Mathematical Journal

- Kodai Math. J.
- Volume 27, Number 1 (2004), 45-56.

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

Aldir Chaves Brasil Jr., Abd\^enago Alves de Barros, and Luis Amancio Machado de Soursa Jr.

#### Abstract

In this work we will consider compact submanifold $M^{n}$ immersed in the Euclidean sphere $S^{n+p}$ with parallel mean curvature vector and we introduce a Schr\"{o}dinger operator $L=-\Delta+V$, where $\Delta$ 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 $\mu_{1}$ of $L$, by showing that either $\mu_{1}=0$ or $\mu_{1}\leq-n(1+H^{2})$. As a consequence we obtain new characterizations of spheres, Clifford tori and Veronese surfaces that extend a work due to Wu \cite{wu} for minimal submanifolds.

#### Article information

**Source**

Kodai Math. J., Volume 27, Number 1 (2004), 45-56.

**Dates**

First available in Project Euclid: 21 May 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.kmj/1085143788

**Digital Object Identifier**

doi:10.2996/kmj/1085143788

**Mathematical Reviews number (MathSciNet)**

MR2042790

**Zentralblatt MATH identifier**

1059.53047

#### Citation

de Barros, Abd\^enago Alves; Brasil Jr., Aldir Chaves; de Soursa Jr., Luis Amancio Machado. A new characterization of submanifolds with parallel mean curvature vector in $S^{n+p}$. Kodai Math. J. 27 (2004), no. 1, 45--56. doi:10.2996/kmj/1085143788. https://projecteuclid.org/euclid.kmj/1085143788