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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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 curvature
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by Isabel Maria da Costa Salavessa PDF
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