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Asymptotic Plateau problem for $3$-convex hypersurface in $\mathbb{H}^5$

Zhenan Sui

Abstract

We prove the existence of a smooth complete $3$-convex hypersurface which satisfies prescribed curvature equation $\prod\limits_{i = 1}^n (H - κ_i) = \big( (n - 1) σ\big)^n$ for $n = 4$ and has prescribed asymptotic boundary $Γ$ at the infinity of hyperbolic space of dimension 5, where $σ\in (0, 1)$ is a constant and $Γ$ is assumed to have nonnegative mean curvature. We introduce Lagrange multiplier method to compute the extreme value of the concavity of $f (κ) = \frac{1}{n - 1} \Big( \prod\limits_{i = 1}^n (H - κ_i) \Big)^{\frac{1}{n}}$ during uniform global curvature estimate.

Asymptotic Plateau problem for $3$-convex hypersurface in $\mathbb{H}^5$

Abstract

We prove the existence of a smooth complete -convex hypersurface which satisfies prescribed curvature equation for and has prescribed asymptotic boundary at the infinity of hyperbolic space of dimension 5, where is a constant and is assumed to have nonnegative mean curvature. We introduce Lagrange multiplier method to compute the extreme value of the concavity of during uniform global curvature estimate.

Paper Structure

This paper contains 18 sections, 12 theorems, 272 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometric quantities on vertical graphs
  4. Fundamental structure conditions
  5. Calculations on hypersurface $\Sigma$
  6. Uniform global curvature estimate
  7. Computing concavity by Lagrange multiplier method
  8. For $i = 1$
  9. For $i = 2$
  10. The case when $0 < \kappa_2 \leq \nu^{n + 1}$.
  11. The case when $\kappa_2 \geq \nu^{n + 1}$.
  12. For $i = 3$
  13. The case when $\kappa_3 \geq - \varphi \kappa_1$.
  14. The case when $\kappa_3 \leq - \varphi \kappa_1$.
  15. For $i = 4$
  16. ...and 3 more sections

Key Result

Theorem 1.1

GS10 Let $\Gamma = \partial \Omega \times \{ 0 \} \subset \mathbb{R}^{n + 1}$ where $\Omega$ is a bounded smooth domain in $\mathbb{R}^n$. Suppose that the Euclidean mean curvature $\mathcal{H}_{\partial \Omega}$ is nonnegative and $\sigma \in (0, 1)$ satisfies $\sigma > \sigma_0$, where $\sigma_0$ (Numerical calculations show $0.3703 < \sigma_0 < 0.3704$.) Under conditions eq2-16--eq2-21, there

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 4.1
  • Remark 4.2
  • Proposition 5.1
  • proof
  • Proposition 5.2
  • proof
  • ...and 10 more