Minimal harmonic graphs and their Lorentzian cousins

Abstract

Motivated by the observation that the only surface which is locally a graph of a harmonic function and is also a minimal surface in E 3 is either a plane or a helicoid, we provide similar characterizations of the elliptic, hyperbolic and parabolic helicoids in L 3 as the nontrivial zero mean curvature surfaces which also satisfy the harmonic equation, the wave equation, and a degenerate equation which is derived from the harmonic equation or the wave equation. This elementary and analytic result shows that the change of the roles of dependent and independent variables may be useful in solving differential equations.

Ruled minimal surfaces in the three-dimensional Heisenberg group

“…If we add (5) and (6), we have f x x + f yy = 0, that is, if a graph of a function z = f (x, y) in ‫ވ‬ 3 satisfies the condition of Theorem 3.2, f must be a harmonic function. This fact is true for the three-dimensional Lorentzian space ‫ތ‬ 3 and is the motivation of [Kim et al 2009b]. We think it is a nontrivial fact and would like to find applications of this fact in future study.…”