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

TL;DR

This paper classifies and describes special constant $p$-mean curvature surfaces in the Heisenberg group $H_1$, focusing on rotationally invariant examples and the geometric significance of an associated energy bound.

Contribution

It provides a complete classification of rotationally invariant constant $p$-mean curvature surfaces in $H_1$ and interprets the energy $E$ within this context.

Findings

Complete description of rotationally invariant constant $p$-mean curvature surfaces.

Identification of the geometric interpretation of the energy $E$ and its lower bound.

New examples of constant $p$-mean curvature surfaces in the Heisenberg group.

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.