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Bernstein-type Theorems for constant mean curvature surfaces in the three-dimensional light cone

Shintaro Akamine, Wonjoo Lee, Seong-Deog Yang

Abstract

We establish Bernstein-type theorems for entire constant mean curvature graphs in the three-dimensional light cone $\mathbb{Q}^3_+$ over the horosphere under the assumption that the Gaussian curvature $K$ is bounded below, by showing that such graphs are horospheres or spheres of $\mathbb{Q}^3_+$.

Bernstein-type Theorems for constant mean curvature surfaces in the three-dimensional light cone

Abstract

We establish Bernstein-type theorems for entire constant mean curvature graphs in the three-dimensional light cone over the horosphere under the assumption that the Gaussian curvature is bounded below, by showing that such graphs are horospheres or spheres of .
Paper Structure (12 sections, 13 theorems, 49 equations, 3 figures)

This paper contains 12 sections, 13 theorems, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hermitian model of $\mathbb{Q}^3_+$
  4. Punctured ball model of $\mathbb{Q}^3_+$
  5. Half space model of $\mathbb{Q}^3_+$ and graphs over a horosphere
  6. Curvatures of graphs in $\mathbb{Q}^3_+$
  7. Spheres, horospheres, and hyperspheres in $\mathbb{Q}^3_+$
  8. Intersection of $\mathbb{Q}^3_+$ with hyperplanes in $\mathbb{L}^4$
  9. Graph functions of the hyperplane sections
  10. Bernstein-type theorems in $\mathbb{Q}^3_+$
  11. Bernstein-type theorem for ZMC graphs in $\mathbb{Q}^3_+$
  12. Bernstein-type Theorem for nonzero CMC graphs in $\mathbb{Q}^3_+$

Key Result

Corollary 1.1

Let $S$ be a spacelike CMC $H$ surface in $\mathbb{Q}^3_+$ such that $\pi|_S$ is a one-to-one correspondence between $S$ and the entire ideal boundary. Then, $H<0$ and $S$ is a sphere.

Figures (3)

  • Figure 1: A sphere in the half space model (left) and in the punctured ball model (right) of $\mathbb{Q}^3_+$.
  • Figure 2: The projection $\pi$ and a graph in the half space model (left) and in the punctured ball model (right).
  • Figure 3: A sphere, two horospheres, and three hyperspheres in $\mathbb{Q}^3_+$, in the half space model (top row) and in the punctured ball model (bottom row). For labels such as S-i in the figure, see Lemma \ref{['Lem:202601171034AM']}.

Theorems & Definitions (28)

  • Corollary 1.1
  • Definition 2.1
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Corollary 3.7
  • ...and 18 more