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Complete minimal hypersurfaces in the hyperbolic space $\mathbb{H}^4$ with vanishing Gauss-Kronecker curvature

T. Hasanis, A. Savas-Halilaj, T. Vlachos

Abstract

We investigate 3-dimensional complete minimal hypersurfaces in the hyperbolic space $\mathbb{H}^{4}$ with Gauss-Kronecker curvature identically zero. More precisely, we give a classification of complete minimal hypersurfaces with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and scalar curvature bounded from below.

Complete minimal hypersurfaces in the hyperbolic space $\mathbb{H}^4$ with vanishing Gauss-Kronecker curvature

Abstract

We investigate 3-dimensional complete minimal hypersurfaces in the hyperbolic space with Gauss-Kronecker curvature identically zero. More precisely, we give a classification of complete minimal hypersurfaces with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and scalar curvature bounded from below.

Paper Structure

This paper contains 4 sections, 9 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. The polar map of stationary surfaces in the de Sitter space
  3. Local theory of minimal hypersurfaces in $\mathbb{H}^{4}$ with zero Gauss-Kronecker curvature
  4. Complete minimal hypersurfaces in $\mathbb{H}^{4}$ with vanishing Gauss-Kronecker curvature

Key Result

Proposition \oldthetheorem

Let $M^{2}$ be a $2$-dimensional Riemannian manifold and $g:M^{2}\rightarrow \mathbb{S}_{1}^{4}$ a stationary isometric immersion. Then

Theorems & Definitions (22)

  • Proposition \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • ...and 12 more