The boundary value problem for maximal hypersurfaces

Abstract

A spacelike hypersurface (condimension 1) in a Lorentzian manifold is called a maximal surface if it extremizes the hypervolume functional. Although maximal surfaces are superficially analogous to minimal hypersurfaces in Riemannian geometry, their properties can be dramatically different, as can be seen from the validity of Bernstein's theorem in all dimensions [Cheng, S.-Y. & Yau, S.-T. (1976) Ann. Math. 104, 407-419]. Here we establish a point of contact between maximal surfaces and minimal surfaces by solving the Dirichlet problem for acausal boundary data but using boundary curvature conditions similar to those of H. Jenkins and J. Serrin [(1968) J. Reine Angew. Math. 229, 170-187].