Graphs with Parallel Mean Curvature

Isabel Maria da Costa Salavessa
1989 Proceedings of the American Mathematical Society  
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
more » ... nonminimal submanifolds of M x R with parallel mean curvature.
doi:10.2307/2047835 fatcat:3omfajmqsrf5xk6tvbvbd7x5au