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Minimal harmonic graphs and their Lorentzian cousins. (English) Zbl 1161.53051

Summary: Motivated by the observation that the only surface which is locally a graph of a harmonic function and is also a minimal surface in \(\mathbb E^3\) is either a plane or a helicoid, we provide similar characterizations of the elliptic, hyperbolic and parabolic helicoids in \(\mathbb L^3\) 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.

MSC:

53C43 Differential geometric aspects of harmonic maps
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53B50 Applications of local differential geometry to the sciences
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References:

[1] Y.W. Kim, S.-E. Koh, H. Shin, S.-D. Yang, Spacelike maximal surfaces, timelike minimal surfaces, and Björling representation formulae, preprint; Y.W. Kim, S.-E. Koh, H. Shin, S.-D. Yang, Spacelike maximal surfaces, timelike minimal surfaces, and Björling representation formulae, preprint
[2] Kim, Y. W.; Yang, S.-D., Prescribing singularities of maximal surfaces via a singular Björling representation formula, J. Geom. Phys., 57, 2167-2177 (2007) · Zbl 1141.53012
[3] Kobayashi, O., Maximal surfaces in the 3-dimensional Minkowski space \(L^3\), Tokyo J. Math., 6, 297-309 (1983) · Zbl 0535.53052
[4] Osserman, R., A Survey of Minimal Surfaces (1986), Dover: Dover New York · Zbl 0209.52901
[5] Weinstein, T., An Introduction to Lorentz Surfaces (1996), de Gruyter: de Gruyter New York · Zbl 0881.53001
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