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