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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
Abstract

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.

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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