A Simons type formula for surfaces with parallel mean curvature

TL;DR

This paper derives a Simons type formula for non-minimal surfaces with parallel mean curvature in certain space forms and uses it to characterize complete surfaces with non-negative Gaussian curvature.

Contribution

It introduces a new Simons type equation for pmc surfaces in product spaces and applies it to classify complete non-minimal pmc surfaces with non-negative Gaussian curvature.

Findings

Derived a Simons type formula for pmc surfaces in $M^n(c)\times\mathbb{R}$

Characterized complete non-minimal pmc surfaces with non-negative Gaussian curvature

Provided new tools for studying the geometry of pmc surfaces in product spaces

Abstract

We prove a Simons type equation for non-minimal surfaces with parallel mean curvature vector (pmc surfaces) in $M^n(c)\times\mathbb{R}$, where $M^n(c)$ is an $n$-dimensional space form. Then, we use this equation in order to characterize complete non-minimal pmc surfaces with non-negative Gaussian curvature.