The prescribed curvature problem for entire hypersurfaces in Minkowski space

Ren, Changyu; Wang, Zhizhang; Xiao, Ling

doi:10.2140/apde.2024.17.1

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The prescribed curvature problem for entire hypersurfaces in Minkowski space

Changyu Ren, Zhizhang Wang and Ling Xiao

Vol. 17 (2024), No. 1, 1–40

DOI: 10.2140/apde.2024.17.1

Abstract

We prove three results in this paper: First, we prove, for a wide class of functions $φ \in C^{2} (𝕊^{n - 1})$ and $ψ (X, ν) \in C^{2} (ℝ^{n + 1} \times ℍ^{n})$ , there exists a unique, entire, strictly convex, spacelike hypersurface $ℳ_{u}$ satisfying $σ_{k} (κ [ℳ_{u}]) = ψ (X, ν)$ and $u (x) \to | x | + φ (x ∕ | x |)$ as $| x | \to \infty$ . Second, when $k = n - 1, n - 2$ , we show the existence and uniqueness of an entire, $k$ -convex, spacelike hypersurface $ℳ_{u}$ satisfying $σ_{k} (κ [ℳ_{u}]) = ψ (x, u (x))$ and $u (x) \to | x | + φ (x ∕ | x |)$ 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 $σ_{k}$ curvature flow equations. Moreover, we prove that the downward translating solitons $ℳ_{u}$ have bounded principal curvatures.

Keywords

prescribed curvature, Minkowski space, downward translating solitons

Mathematical Subject Classification

Primary: 53C42

Secondary: 35J60, 49Q10, 53C50

Milestones

Received: 12 July 2020

Revised: 18 May 2022

Accepted: 11 July 2022

Published: 5 February 2024

Authors

Changyu Ren
School of Mathematical Sciences Jilin University Changchun China

Zhizhang Wang
School of Mathematical Sciences Fudan University Shanghai China

Ling Xiao
University of Connecticut Storrs, CT United States

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Vol. 17, No. 1, 2024