On Area-Stationary Surfaces in Certain Neutral Kaehler 4-Manifolds

Authors

Brendan Guilfoyle, School of Science, Technology, Engineering and Mathematics, Institute of Technology, Tralee, Clash, Tralee, Co. Kerry, Ireland.Follow
Wilhelm Klingenberg, Department of Mathematical Sciences, University of Durham, Durham, DH1 3LE, UK

ORCID

0000-0002-7437-5046

Document Type

Article

Disciplines

Mathematics | Physical Sciences and Mathematics

Publication Details

Beiträge zur Algebra und Geometrie, 49 (2). © 2008 Springer

Abstract

We study surfaces in TN that are area-stationary with respect to a neutral Kaehler metric constructed on TN from a riemannian metric g on N. We show that holomorphic curves in TN are area-stationary, while lagrangian surfaces that are area-stationary are also holomorphic and hence totally null. 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→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 R3, and we exhibit a two parameter family of area-stationary tori that are neither holomorphic nor lagrangian.

Recommended Citation

Guilfoyle, Brendan & Klingenberg, Wilhelm (2008). Area-stationary surfaces in neutral Kähler 4-manifolds. Beiträge zur Algebra und Geometrie 49 (2): 481-490.