Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 49, No. 2, pp. 481-490 (2008)

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Area-stationary surfaces in neutral Kähler 4-manifolds

Brendan Guilfoyle and Wilhelm Klingenberg

Department of Mathematics and Computing, Institute of Technology, Tralee Clash, Tralee, Co. Kerry, Ireland, e-mail:, Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, United Kingdom, e-mail:

Abstract: We study surfaces in TN that are area-stationary with respect to a neutral Kähler metric constructed on TN from a Riemannian metric g on N. We show that holomorphic curves in TN are area-stationary. However, in general, area stationary surfaces are not holomorphic. We prove this by constructing counter-examples. In the case where g is rotationally symmetric, we find all area stationary surfaces that arise as graphs of sections of the bundle TN$\rightarrow$N and that are rotationally symmetric. When (N,g) is the round 2-sphere, TN can be identified with the space of oriented affine lines in ${\Bbb{R}}^3$, and we exhibit a two parameter family of area-stationary tori that are neither holomorphic nor Lagrangian.

Keywords: maximal surface, mean curvature, neutral Kähler

Classification (MSC2000): 53B30; 53A25

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Electronic version published on: 18 Sep 2008. This page was last modified: 28 Jan 2013.

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