ENTIRE SPACELIKE HYPERSURFACES OF CONSTANT MEAN CURVATURE IN MINKOWSKI SPAC

We describe and classify the complete constant mean curvature spacelike hypersurfaces in Minkowski Space of arbitrary dimension. A method is developed for constructing these hypersurfaces. It is used in building counterexamples to the conjecture that the only such hypersurfaces are the hyperboloids. New curvature estimates are given for constant mean curvature hypersurfaces.;Spacelike hypersurfaces in Minkowski Space are given by graphs of functions whose gradient has norm less than one. We treat functions that satisfy the Minkowski Space Constant Mean Curvature Equation over all of Euclidean Space. This is a quasilinear partial differential equation which is elliptic only for spacelike graphs. The projective boundary of an entire solution is defined to be the limiting hypersurface obtained by homothetically contracting the surrounding Minkowski Space to a point. We show that the projective boundaries gotten from entire solutions belong to the set of convex positively homogeneous functions whose gradient has norm one wherever defined.;On the other hand, given a positively homogeneous convex function whose gradient has length one when defined, then there exists a spacelike constant mean curvature entire hypersurface whose projective boundary equals the given function.;The method of proof is approximation by a sequence of solutions. Entire sub- and supersolutions are constructed by exploiting the structure of the boundary data and the availability of explicit solutions. After the necessary gradient bounds are obtained, a Dirichlet problem is solved for the Minkowski Space Constant Mean Curvature Equation in an expanding sequence of domains with boundary data between the entire barriers. That this approximating sequence converges to a solution is deduced from uniform third derivative bounds on the functions defining the approximates. These in turn are obtained from intrinsic interior estimates on the second fundamental form and it's covariant derivatives. Application of these estimates is possible after establishing that the intrinsic radii of the approximating solutions grow sufficiently quickly.;Examples are also given to show that there may be several nonisometric solutions with the same projective boundary.;In summary, up to equivalence of projective boundaries, the entire spacelike hypersurfaces of constant mean curvature in Minkowski Space coincide with the convex positively homogeneous functions whose gradients have norm one wherever defined.