A characterization of constant \(p\)-mean curvature surfaces in the Heisenberg group \(H_1\)

In Euclidean \(3\)-space, it is well known that the Sine-Gordon equation was considered in the nineteenth century in the course of investigations of surfaces of constant Gaussian curvature \(K=-1\). Such a surface can be constructed from a solution to the Sine-Gordon equation, and vice versa. With t...

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Bibliographic Details
Published in:arXiv.org 2022-05
Main Authors: Hung-Lin, Chiu, Hsiao-Fan, Liu
Format: Article
Language:English
Subjects:
Curvature
Euclidean geometry
Minimal surfaces
Theorems
Online Access:Get full text
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Summary:In Euclidean \(3\)-space, it is well known that the Sine-Gordon equation was considered in the nineteenth century in the course of investigations of surfaces of constant Gaussian curvature \(K=-1\). Such a surface can be constructed from a solution to the Sine-Gordon equation, and vice versa. With this as motivation, employing the fundamental theorem of surfaces in the Heisenberg group \(H_{1}\), we show in this paper that the existence of a constant \(p\)-mean curvature surface (without singular points) is equivalent to the existence of a solution to a nonlinear second-order ODE (1.2), which is a kind of {\bf Li\'{e}nard equations}. Therefore, we turn to investigate this equation. It is a surprise that we give a complete set of solutions to (1.2) (or (1.5)), and hence use the types of the solution to divide constant \(p\)-mean curvature surfaces into several classes. As a result, after a kind of normalization, we obtain a representation of constant \(p\)-mean curvature surfaces and classify further all constant \(p\)-mean curvature surfaces. In Section 9, we provide an approach to construct \(p\)-minimal surfaces. It turns out that, in some sense, generic \(p\)-minimal surfaces can be constructed via this approach. Finally, as a derivation, we recover the Bernstein-type theorem which was first shown in [3] (or see [7,8]).
ISSN:2331-8422