Flat Lorentz Surfaces in Anti-de Sitter 3-space and Gravitational Instantons

In this paper, we study flat Lorentz surfaces in anti-de Sitter 3-space H-1(3)(-1) in terms of the second conformal structure. Those flat Lorentz surfaces can be represented in terms of a Lorentz holomorphic and a Lorentz anti-holomorphic data similarly to WeierstraB representation formula. An analogue of hyperbolic GauB map is considered for timelike surfaces in H-1(3)(-1) and the relationship between the conformality (or the holomorphicity) of hyperbolic GauB map and the flatness of a Lorentz surface is discussed. It is shown that flat Lorentz surfaces in H-1(3)(-1) are associated with a hyperbolic Monge-Ampere ere equation. It is also known that Monge-Ampere equation may be regarded as a 2-dimensional reduction of the Einstein\u27s field equation. Using this connection, we construct a class of anti-self-dual gravitational instantons from flat Lorentz surfaces in H-1(3)(-1)