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Graphs with Parallel Mean Curvature

1989
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Proceedings of the American Mathematical Society
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We prove that if the graph Tf = {(x,f(x)): x e M} of a map /: (M, g) -> (TV, h) between Riemannian manifolds is a submanifold of (M x N,gxh) with parallel mean curvature H , then on a compact domain D C M , \\H\\ is bounded from above by ^ ffiff . In particular, ry 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

doi:10.2307/2047835
fatcat:3omfajmqsrf5xk6tvbvbd7x5au