Graphs with parallel mean curvature

Abstract

We prove that if the graph Γ f = { ( x , f ( x ) ) : x ∈ M } {\Gamma _f} = \left \{ {\left ( {x,f\left ( x \right )} \right ):x \in M} \right \} of a map f : ( M , g ) → ( N , h ) f:\left ( {M,g} \right ) \to \left ( {N,h} \right ) between Riemannian manifolds is a submanifold of ( M × N , g × h ) \left ( {M \times N,g \times h} \right ) with parallel mean curvature H H , then on a compact domain D ⊂ M D \subset M , ‖ H ‖ \left \| H \right \| is bounded from above by 1 m A ( ∂ D ) V ( D ) \frac {1}{m}\frac {{A\left ( {\partial D} \right )}}{{V\left ( D \right )}} . In particular, Γ f {\Gamma _f} is minimal provided M M is compact, or noncompact with zero Cheeger constant. Moreover, if M M is the m m -hyperbolic space—thus with nonzero Cheeger constant—then there exist real-valued functions the graphs of which are nonminimal submanifolds of M × R M \times \mathbb {R} with parallel mean curvature.