Communications in Analysis and Geometry

Volume 30 (2022)

Number 8

Stochastically complete submanifolds with parallel mean curvature vector field in a Riemannian space form

Pages: 1793 – 1809

DOI: https://dx.doi.org/10.4310/CAG.2022.v30.n8.a4

Authors

Henrique F. de Lima (Departamento de Matemática Universidade Federal de Campina Grande, Campina Grande, Paraíba, Brazil)

Fábio R. dos Santos (Departamento de Matemática, Universidade Federal de Pernambuco, Recife, Pernambuco, Brazil)

Abstract

In this paper, we deal with stochastically complete submanifolds $M^n$ immersed with nonzero parallel mean curvature vector field in a Riemannian space form $\mathbb{Q}^{n+p}_c$ of constant sectional curvature $c \in {\lbrace -1, 0, 1 \rbrace}$. In this setting, we use the weak Omori–Yau maximum principle jointly with a suitable Simons type formula in order to show that either such a submanifold $M^n$ must be totally umbilical or it holds a sharp estimate for the norm of its total umbilicity tensor, with equality if and only if the submanifold is isometric to an open piece of a hyperbolic cylinder $\mathbb{H}^1 {\left( -\sqrt{1+r^2} \right)} \times \mathbb{S}^{n-1} (r)$ when $c=-1$, a circular cylinder $\mathbb{R} \times S^{n-1} (r)$, when $c=0$, and a Clifford torus $\mathbb{S}^1 {\left( 1-r^2 \right)} \times \mathbb{S}^{n-1} (r)$, when $c=0$.

Received 11 July 2016

Accepted 11 March 2020

Published 13 July 2023