The prescribed curvature problem for entire hypersurfaces in Minkowski space

Changyu Ren; Zhizhang Wang; Ling Xiao

Paper · Jul 9, 2020

The prescribed curvature problem for entire hypersurfaces in Minkowski space

Changyu Ren, Zhizhang Wang, Ling Xiao

math.DGWider flowsarXiv:2007.04493

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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 $u (x) \to ∣ x ∣ + φ (\frac{x}{∣ x ∣})$ as $∣ x ∣ \to \infty.$ Second, when $k = n - 1, n - 2,$ we show the existence and uniqueness of entire, $k$ -convex, spacelike hypersurface $M_{u}$ satisfying $σ_{k} (κ [M_{u}]) = ψ (x, u (x))$ and $u (x) \to ∣ x ∣ + φ (\frac{x}{∣ x ∣})$ 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.

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Submitted: Jul 9, 2020
Subjects: math.DG, math.AP
Journal: Analysis & PDE 17 (2024) 1-40
DOI: 10.2140/apde.2024.17.1

Other curvature flows Self-shrinkers & translators Stability & uniqueness Physics

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The prescribed curvature problem for entire hypersurfaces in Minkowski space

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