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