On area-stationary surfaces in certain neutral Kaehler 4-manifolds

Details On area-stationary surfaces in certain neutral Kaehler 4-manifolds On area-stationary surfaces in certain neutral Kaehler 4-manifolds

2006-11-22

arxiv.org/abs/math/0611707v1

Beitraege Algebra Geom. 49 (2008) 481-490.

Mathematics

Differential Geometry

8 pages, AMS-LATEX, no figures

Scientific paper

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$\to$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.

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