Vol. 288, No. 1, 2017

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ISSN: 0030-8730
Geometric properties of level curves of harmonic functions and minimal graphs in 2-dimensional space forms

Jinju Xu and Wei Zhang

Vol. 288 (2017), No. 1, 217–239
DOI: 10.2140/pjm.2017.288.217
Abstract

We study the geometric properties of level curves of harmonic functions and minimal graphs in 2-dimensional space forms using the maximum principle. More precisely, we find two auxiliary functions which consist of tangential derivatives of the curvature of level curves and the norms of the gradient of the solution functions. Then we prove that they satisfy certain elliptic partial differential equations.

Keywords
level curves, harmonic functions, minimal graphs, space forms
Mathematical Subject Classification 2010
Primary: 35B45, 35B50
Secondary: 35J05, 35J93
Milestones
Received: 25 May 2016
Revised: 24 September 2016
Accepted: 19 November 2016
Published: 8 April 2017
Authors
Jinju Xu
Department of Mathematics
Shanghai Normal University
Shanghai 200234
China
Wei Zhang
School of Mathematics & Statistics
Lanzhou University
Gansu 730000
China