A characterization of $p$-minimal surfaces in the Heisenberg group $H_1$

Hung-Lin Chiu, Hsiao-Fan Liu

Published 2021-01-28Version 1

In Euclidean $3$-space, it is well known that the Sine-Gordon equation was considered in the nineteenth century in the course of investigation 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, by means of 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, 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 such ODEs in the $p$-minimal case, and hence use the types of the solution to divide $p$-minimal surfaces into several classes. As a result, we obtain a representation of $p$-minimal surfaces and classify further all $p$-minimal 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 2005.