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