Minimal harmonic graphs and their Lorentzian cousins

Young Wook Kim; Hyung Yong Lee; Seong Deog Yang

doi:10.1016/j.jmaa.2008.12.025

Minimal harmonic graphs and their Lorentzian cousins

Young Wook Kim, Hyung Yong Lee, Seong Deog Yang

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

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 E³ is either a plane or a helicoid, we provide similar characterizations of the elliptic, hyperbolic and parabolic helicoids in L³ 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.

Original language	English
Pages (from-to)	666-670
Number of pages	5
Journal	Journal of Mathematical Analysis and Applications
Volume	353
Issue number	2
DOIs	https://doi.org/10.1016/j.jmaa.2008.12.025
Publication status	Published - 2009 May 15

Keywords

Harmonic graph
Minimal surfaces
Wave graph
Zero mean curvature surfaces

ASJC Scopus subject areas

Analysis
Applied Mathematics

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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 E3 is either a plane or a helicoid, we provide similar characterizations of the elliptic, hyperbolic and parabolic helicoids in L3 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.",

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note = "Funding Information: We thank J. Inoguchi for pointing out Kobayashi{\textquoteright}s result. The first and the last authors were supported in part by Korea University research grants. Copyright: Copyright 2009 Elsevier B.V., All rights reserved.",

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AU - Kim, Young Wook

AU - Lee, Hyung Yong

AU - Yang, Seong Deog

N1 - Funding Information: We thank J. Inoguchi for pointing out Kobayashi’s result. The first and the last authors were supported in part by Korea University research grants. Copyright: Copyright 2009 Elsevier B.V., All rights reserved.

PY - 2009/5/15

Y1 - 2009/5/15

N2 - Motivated by the observation that the only surface which is locally a graph of a harmonic function and is also a minimal surface in E3 is either a plane or a helicoid, we provide similar characterizations of the elliptic, hyperbolic and parabolic helicoids in L3 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.

AB - Motivated by the observation that the only surface which is locally a graph of a harmonic function and is also a minimal surface in E3 is either a plane or a helicoid, we provide similar characterizations of the elliptic, hyperbolic and parabolic helicoids in L3 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.

KW - Harmonic graph

KW - Minimal surfaces

KW - Wave graph

KW - Zero mean curvature surfaces

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Minimal harmonic graphs and their Lorentzian cousins

Abstract

Keywords

ASJC Scopus subject areas

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