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On Invariants of Constant $p$-Mean Curvature Surfaces in the Heisenberg Group $H_1$

Hung-Lin Chiu, Sin-Hua Lai, Hsiao-Fan Liu

TL;DR

The paper addresses the classification and explicit construction of constant $p$-mean curvature surfaces in the Heisenberg group $H_1$, with emphasis on rotationally invariant examples and a clear energy interpretation. It advances the framework by solving the Codazzi-like equation for the $\alpha$-invariant, normalizing the induced metric to obtain complete invariants $(\zeta_1,\zeta_2)$, and analyzing singular sets; it further develops a deformation-based method to generate new constant $p$-mean curvature surfaces from the Pansu sphere. The main contributions include a complete description of rotationally invariant constant $p$-mean curvature surfaces, explicit invariant and energy formulas, a normalization scheme with unique invariant data, and construction techniques that yield broad families including p-minimal and Pansu-sphere–like surfaces. These results deepen the understanding of submanifold geometry in $H_1$, connect to classical examples such as the Pansu sphere, and provide concrete tools for exploring CR and sub-Riemannian geometry in low dimensions.

Abstract

One primary objective in submanifold geometry is to discover fascinating and significant classical examples of $H_1$. In this paper which relies on the theory we established in [Adv. Math. 405 (2022), 08514, 50 pages, arXiv:2101.11780] and utilizing the approach we provided for constructing constant $p$-mean curvature surfaces, we have identified intriguing examples of such surfaces. Notably, we present a complete description of rotationally invariant surfaces of constant $p$-mean curvature and shed light on the geometric interpretation of the energy $E$ with a lower bound.

On Invariants of Constant $p$-Mean Curvature Surfaces in the Heisenberg Group $H_1$

TL;DR

The paper addresses the classification and explicit construction of constant -mean curvature surfaces in the Heisenberg group , with emphasis on rotationally invariant examples and a clear energy interpretation. It advances the framework by solving the Codazzi-like equation for the -invariant, normalizing the induced metric to obtain complete invariants , and analyzing singular sets; it further develops a deformation-based method to generate new constant -mean curvature surfaces from the Pansu sphere. The main contributions include a complete description of rotationally invariant constant -mean curvature surfaces, explicit invariant and energy formulas, a normalization scheme with unique invariant data, and construction techniques that yield broad families including p-minimal and Pansu-sphere–like surfaces. These results deepen the understanding of submanifold geometry in , connect to classical examples such as the Pansu sphere, and provide concrete tools for exploring CR and sub-Riemannian geometry in low dimensions.

Abstract

One primary objective in submanifold geometry is to discover fascinating and significant classical examples of . In this paper which relies on the theory we established in [Adv. Math. 405 (2022), 08514, 50 pages, arXiv:2101.11780] and utilizing the approach we provided for constructing constant -mean curvature surfaces, we have identified intriguing examples of such surfaces. Notably, we present a complete description of rotationally invariant surfaces of constant -mean curvature and shed light on the geometric interpretation of the energy with a lower bound.
Paper Structure (19 sections, 15 theorems, 125 equations, 1 figure, 2 tables)

This paper contains 19 sections, 15 theorems, 125 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Solutions to the Codazzi-like equation
  3. The Pansu sphere
  4. The normalization
  5. The structure of the singular sets
  6. Rotationally invariant surfaces in $\boldsymbol{ H_1}$
  7. The computation of $\boldsymbol{H}$, $\boldsymbol{\alpha}$, $\boldsymbol{a}$ and $\boldsymbol{b}$
  8. Another understanding of energy $\boldsymbol{ E}$
  9. The Coddazi-like equation
  10. The relation between $\boldsymbol{k}$ and $\boldsymbol{E}$
  11. Horizontal generating curves for $\boldsymbol{c\neq 0}$
  12. The invariants $\boldsymbol{\zeta_{1}}$ and $\boldsymbol{\zeta_{2}}$ for surfaces with $\boldsymbol{c\neq 0}$
  13. The allowed values of $\boldsymbol{ k}$ and $\boldsymbol{E}$ with $\boldsymbol{c=0}$
  14. The invariants $\boldsymbol{\zeta_{1}}$ and $\boldsymbol{\zeta_{2}}$ for surfaces with $\boldsymbol{c=0}$ and $\boldsymbol{E\neq 0}$
  15. The construction of constant $\boldsymbol{ p}$-mean curvature surfaces
  16. ...and 4 more sections

Key Result

Theorem A

A curve $\gamma=(x,t)$ is the generating curve of a rotationally invariant surface $\Sigma$ in $H_{1}$ with $H=c\neq 0$ if and only if $\gamma=(x,t)$ is defined by $x^{2}=\frac{k}{c^2}+r\cos{(c\tilde{s})}$, $t=-\frac{\tilde{s}}{c}-\frac{r}{2}\sin{(c\tilde{s})}$, up to a constant, for some horizontal In addition, we have $k=2cE+2$. If $r=0$, then $\Sigma$ is a cylinder. If $r\neq 0$, then, in terms

Figures (1)

  • Figure 1: Direction field for $c=1.5$.

Theorems & Definitions (22)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem 2.1: ChiuH/LiuH:2022
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4: Pansu sphere
  • Proposition 2.5
  • Theorem 2.6
  • Theorem 2.7
  • ...and 12 more