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# 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

#### 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 $c=0$.

The first author is partially supported by CNPq, Brazil, grant 301970/2019-0.

The second author is partially supported by CNPq, Brazil, grants 431976/2018-0 and 311124/2021-6 and Propesqi (UFPE).

Received 11 July 2016

Accepted 11 March 2020

Published 13 July 2023