The prescribed curvature problem for entire hypersurfaces in Minkowski space

TL;DR

This paper establishes existence and uniqueness results for entire hypersurfaces in Minkowski space with prescribed curvature functions, including special cases and translating solitons, advancing the understanding of geometric PDEs in Lorentzian geometry.

Contribution

It proves new existence and uniqueness theorems for entire hypersurfaces with prescribed curvature in Minkowski space, including special cases and solitons, with bounded principal curvatures.

Findings

Existence and uniqueness of hypersurfaces with prescribed $\sigma_k$ curvature.

Construction of downward translating solitons with prescribed asymptotics.

Bounded principal curvatures of the constructed solitons.

Abstract

We prove three results in this paper. First, we prove for a wide class of functions $\varphi\in C^2(\mathbb{S}^{n-1})$ 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 $u(x)\rightarrow |x|+\varphi\left(\frac{x}{|x|}\right)$ 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 $\sigma_k(\kappa[M_u])=\psi(x, u(x))$ and $u(x)\rightarrow |x|+\varphi\left(\frac{x}{|x|}\right)$ 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 $\sigma_k$ curvature flow equations. Moreover, we prove that the downward translating solitons $M_u$ have bounded principal curvatures.