The prescribed curvature problem for entire hypersurfaces in Minkowski space

Abstract

We prove three results in this paper. First, we prove for a wide class of functions $\phi \in C^2({\mathbb{S}}^{n-1})$ and $\psi (X,\nu )\in C^2({\mathbb{R}}^{n+1}\times {\mathbb{H}}^n),$ there exists a unique, entire, strictly convex, spacelike hypersurface $M_u$ satisfying ${\sigma }_k(\kappa [M_u])=\psi (X,\nu )$ and $u(x)\to |x|+\phi \left(\frac{x}{|x|}\right)$ as $|x|\to \infty .$ Second, when $k=n-1,n-2,$ we show the existence and uniqueness of entire, $k$-convex, spacelike hypersurface $M_u$ satisfying ${\sigma }_k(\kappa [M_u])=\psi (x,u(x))$ and $u(x)\to |x|+\phi \left(\frac{x}{|x|}\right)$ as $|x|\to \infty .$ Last, we obtain the existence and uniqueness of entire, strictly convex, downward translating solitons $M_u$ with prescribed asymptotic behavior at infinity for ${\sigma }_k$ curvature flow equations. Moreover, we prove that the downward translating solitons $M_u$ have bounded principal curvatures.