On Area-stationary Surfaces in Certain Neutral Kähler 4-manifolds Brendan Guilfoyle and Wilhelm Klingenberg

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, 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 R, and we exhibit a two parameter family of area-stationary tori that are neither holomorphic nor lagrangian. One-dimensional submanifolds of neutral Kähler four-manifolds have been studied recently in the context of twistor theory and integrable systems (cf. [2] and references therein). For example, quotienting out by the integral curves of non-null or null Killing vector fields leads to Einstein-Weyl three-manifolds or projective surfaces, respectively. In the case of two-dimensional submanifolds in neutral Kähler four-manifolds, the objects of study in this paper, the situation is more complex. In particular, the metric induced on such a submanifold by the neutral metric can be positive or negative definite, Lorentz or degenerate, with two possible degrees of degeneracy. Moreover, while the geometry of surfaces in positive definite Kähler four-manifolds is well developed, particularly in the Kähler-Einstein case [5], for indefinite metrics many of these results do not hold. The main purpose of this paper then is two-fold: to investigate the geometric properties of two-dimensional submanifolds of a class of neutral Kähler fourmanifolds and, by so doing, to illustrate the differences between the neutral and Hermitian cases. The particular class of neutral Kähler structures we consider have recently been studied on TN, the total space of the tangent bundle to a riemannian two-manifold (N,g) [3] [4]. This construction is motivated by the neutral Kähler metric on the space of oriented lines in R and on the space of time-like lines in R. Aside from the signature, these Kähler four-manifolds differ from the more commonly studied Kähler four-manifolds in a number of crucial ways: they are non-compact and are Kähler-Einstein only in the case when (N,g) is flat. They are, however, scalar flat, and the symplectic structure is exact. Date: 10th November, 2006. 1991 Mathematics Subject Classification. Primary: 53B30; Secondary: 53A25.