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The prescribed curvature problem for entire hypersurfaces in Minkowski space

Changyu Ren, Zhizhang Wang, Ling Xiao

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 $ψ(X, ν)\in C^2(\mathbb{R}^{n+1}\times\mathbb{H}^n),$ there exists a unique, entire, strictly convex, spacelike hypersurface $M_u$ satisfying $σ_k(κ[M_u])=ψ(X, ν)$ and $u(x)\rightarrow |x|+\varphi\left(\frac{x}{|x|}\right)$ 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 $σ_k(κ[M_u])=ψ(x, u(x))$ 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 $σ_k$ curvature flow equations. Moreover, we prove that the downward translating solitons $M_u$ have bounded principal curvatures.

The prescribed curvature problem for entire hypersurfaces in Minkowski space

Abstract

We prove three results in this paper. First, we prove for a wide class of functions and there exists a unique, entire, strictly convex, spacelike hypersurface satisfying and as Second, when we show the existence and uniqueness of entire, -convex, spacelike hypersurface satisfying and as Last, we obtain the existence and uniqueness of entire, strictly convex, downward translating solitons with prescribed asymptotic behavior at infinity for curvature flow equations. Moreover, we prove that the downward translating solitons have bounded principal curvatures.

Paper Structure

This paper contains 21 sections, 27 theorems, 296 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Vertical graphs in $\mathbb{R}^{n, 1}$
  4. The Gauss map
  5. Legendre transform
  6. The Dirichlet problem
  7. Dirichilet problem for $1\leqslant k\leqslant n$
  8. Dirichilet problem for $k=n-1, n-2$
  9. $C^1$ estimate for equation \ref{['Dirichlet curvature']}
  10. $C^2$ boundary estimates for equation \ref{['Dirichlet curvature']}
  11. $C^2$ global estimate for equation \ref{['Dirichlet curvature']}
  12. The local estimates
  13. Local $C^1$ estimates
  14. The Pogorelov type local $C^2$ estimates
  15. The prescribed curvature problem
  16. ...and 6 more sections

Key Result

Theorem 1

Suppose $\varphi$ is a $C^2$ function defined on $\mathbb{S}^{n-1},$ i.e., $\varphi\in C^2(\mathbb{S}^{n-1})$, $\psi(X,\nu)\in C^2(\mathbb{R}^{n+1}\times \mathbb{H}^n)$ is a positive function, and $c_1\geqslant\psi(X,\nu)\geqslant c_2$ for some positive constants $c_1,c_2$. We further assume that $\

Theorems & Definitions (50)

  • Theorem 1
  • Remark 2
  • Remark 3
  • Definition 4
  • Theorem 5
  • Theorem 6
  • Remark 7
  • Lemma 8
  • Lemma 9
  • proof
  • ...and 40 more