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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

We prove three results in this paper: First, we prove, for a wide class of functions φ C2(𝕊n1) and ψ(X,ν) C2(n+1× n), there exists a unique, entire, strictly convex, spacelike hypersurface u satisfying σk(κ[u]) = ψ(X,ν) and u(x) |x| + φ(x|x|) as |x|. Second, when k = n1,n2, we show the existence and uniqueness of an entire, k-convex, spacelike hypersurface u satisfying σk(κ[u]) = ψ(x,u(x)) and u(x) |x| + φ(x|x|) as |x|. Last, we obtain the existence and uniqueness of entire, strictly convex, downward translating solitons u with prescribed asymptotic behavior at infinity for σk curvature flow equations. Moreover, we prove that the downward translating solitons u have bounded principal curvatures.

prescribed curvature, Minkowski space, downward translating solitons
Mathematical Subject Classification
Primary: 53C42
Secondary: 35J60, 49Q10, 53C50
Received: 12 July 2020
Revised: 18 May 2022
Accepted: 11 July 2022
Published: 5 February 2024
Changyu Ren
School of Mathematical Sciences
Jilin University
Zhizhang Wang
School of Mathematical Sciences
Fudan University
Ling Xiao
University of Connecticut
Storrs, CT
United States

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