Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Duality of zero mean curvature surfaces in the Lorentzian Heisenberg group

Sai Rasmi Ranjan Mohanty, Priyank Vasu

Abstract

We study a transformation surface associated with a zero mean curvature surface in the three-dimensional Heisenberg group with respect to two left-invariant semi-Riemannian metrics. We investigate the duality and prove that the transformation surface also has zero mean curvature. Furthermore, we derive the Sym formula for the dual surface in both metric cases.

Duality of zero mean curvature surfaces in the Lorentzian Heisenberg group

Abstract

We study a transformation surface associated with a zero mean curvature surface in the three-dimensional Heisenberg group with respect to two left-invariant semi-Riemannian metrics. We investigate the duality and prove that the transformation surface also has zero mean curvature. Furthermore, we derive the Sym formula for the dual surface in both metric cases.

Paper Structure

This paper contains 10 sections, 9 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Dual maximal surfaces in Nil_1^3 and the Sym formula
  3. Preliminaries for maximal surfaces
  4. Dual maximal surface
  5. Sym formula for dual maximal surface
  6. Singularities for maximal surfaces
  7. Dual Timelike minimal surfaces in $\operatorname{Nil_3}$ and its Sym formula
  8. Preliminaries for timelike minimal surfaces
  9. Dual timelike minimal surface
  10. The Sym formula for dual timelike minimal surface

Key Result

Theorem 2.1

BranderKobayashi2025 Let $f:\mathbb{D}\longrightarrow\mathrm{Nil_1^3}$ be a conformal spacelike immersion and $\alpha_\lambda$ the $1$-form defined in eq:maure_Cartan_H_nonzero and $g$ is the normal Gauss map. Then the following statements are equivalent:

Theorems & Definitions (17)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Theorem 2.6
  • Remark 3.1
  • ...and 7 more