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摘要
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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, employing the fundamental theorem of surfaces in the Heisenberg group H1, 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 Li ́enard 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)) 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 [3] (or see [6, 7]). |