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 $φ\in C^2(\mathbb{S}^{n-1})$ and $ψ(X, ν)\in C^2(\mathbb{R}^{n+1}\times\mathbb{H}^n),$ there exists a unique, entire, strictly convex, spacelike hypersurface $M_u$ satisfying $σ_k(κ[M_u])=ψ(X, ν)$ and $u(x)\rightarrow |x|+φ\left(\frac{x}{|x|}\right)$ as $|x|\rightarrow\infty.$ Second, when $k=n-1, n-2,$ we show the existence and uniqueness of entire, $k$-convex, spacelike hypersurface $M_u$ satisfying $σ_k(κ[M_u])=ψ(x, u(x))$ and $u(x)\rightarrow |x|+φ\left(\frac{x}{|x|}\right)$ as $|x|\rightarrow\infty.$ Last, we obtain the existence and uniqueness of entire, strictly convex, downward translating solitons $M_u$ with prescribed asymptotic behavior at infinity for $σ_k$ curvature flow equations. Moreover, we prove that the downward translating solitons $M_u$ have bounded principal curvatures.