Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



Graphs with parallel mean curvature

Author: Isabel Maria da Costa Salavessa
Journal: Proc. Amer. Math. Soc. 107 (1989), 449-458
MSC: Primary 53C40; Secondary 53C42
MathSciNet review: 965247
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53C40, 53C42

Retrieve articles in all journals with MSC: 53C40, 53C42

Additional Information

Article copyright: © Copyright 1989 American Mathematical Society