In joint work with F. Bonsante and A. Seppi, we solve a
Dirichlet-type problem for entire constant mean curvature hypersurfaces in
Minkowski n+1-space, proving that such surfaces are essentially in bijection
with lower semicontinuous functions on the n-1-sphere. This builds off of
existence theorems by Treibergs and Choi-Treibergs, which themselves rely on
the foundational work of Cheng and Yau. I'll present their maximum principle
argument as well the extra tool that leads to our complete existence and
uniqueness theorem. Time permitting, I'll compare with the analogous problem
of constant Gaussian curvature and present a new result on their intrinsic
geometry.