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Proceedings of the American Mathematical Society

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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
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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\Vert H \right\Vert$ 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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DOI: https://doi.org/10.1090/S0002-9939-1989-0965247-X
Article copyright: © Copyright 1989 American Mathematical Society