# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Graphs with parallel mean curvatureHTML articles powered by AMS MathViewer

by Isabel Maria da Costa Salavessa
Proc. Amer. Math. Soc. 107 (1989), 449-458 Request permission

## Abstract:

We prove that if the graph ${\Gamma _f} = \left \{ {\left ( {x,f\left ( x \right )} \right ):x \in M} \right \}$ of a map $f:\left ( {M,g} \right ) \to \left ( {N,h} \right )$ between Riemannian manifolds is a submanifold of $\left ( {M \times N,g \times h} \right )$ with parallel mean curvature $H$, then on a compact domain $D \subset M$, $\left \| H \right \|$ is bounded from above by $\frac {1}{m}\frac {{A\left ( {\partial D} \right )}}{{V\left ( D \right )}}$. In particular, ${\Gamma _f}$ is minimal provided $M$ is compact, or noncompact with zero Cheeger constant. Moreover, if $M$ is the $m$-hyperbolic space—thus with nonzero Cheeger constant—then there exist real-valued functions the graphs of which are nonminimal submanifolds of $M \times \mathbb {R}$ with parallel mean curvature.
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Additional Information
• © Copyright 1989 American Mathematical Society
• Journal: Proc. Amer. Math. Soc. 107 (1989), 449-458
• MSC: Primary 53C40; Secondary 53C42
• DOI: https://doi.org/10.1090/S0002-9939-1989-0965247-X
• MathSciNet review: 965247