Moduli Spaces of Einstein Metrics on 4-manifolds

In this note, we announce some results showing unexpected similarities between the moduli spaces of constant curvature metrics on 2-manifolds (the Riemann moduli space) and moduli spaces of Einstein metrics on 4manifolds. Let J? denote the moduli space of Einstein metrics of volume 1 on a compact, orientable 4-manifold M. If J£\ denotes the space of smooth Riemannian metrics of volume 1 on M, endowed with a suitable smooth topology, then the diffeomorphism group 3 acts smoothly on «/#! and W is the subspace of Jf\ (3 consisting of (equivalence classes) of Riemannian metrics g satisfying the Einstein condition E(g) = Ric(g) (X/4)g = 0; here Ric denotes the Ricci curvature and X the scalar curvature. It is well known that A is a constant for Einstein metrics in dimension > 3. Concerning the coarse structure of I?, it is known [6] that % consists of countably many components ^ , each of which is locally the quotient of a finite-dimensional real-analytic Hausdorff variety by a compact group action. The scalar curvature variable X: & —• R is constant on fê. There is a natural Riemannian metric on J£\, the L metric, defined as follows: for a,/? symmetric 2-tensors in TgJ[\ = S (M), let (a,/?) = fM(a(x),(i(x))gdvg(x), where ( , )g is the inner product on S (M) induced by g and dvg is the volume form given by g. This induces a Riemannian metric on J[\ j3 and thus a metric on the components fê (since ^ is real-analytically path connected). Note however that the L metric on Jf\j3J is never locally complete (i.e. small metric balls are not complete). In dimension 2, Einstein metrics are naturally considered to be metrics of constant scalar curvature. Thus, % is exactly the space of constant curvature metrics, or equivalently, the space of complex structures, on a closed oriented surface M. Of course, this has been widely studied, see e.g. [5]. In this case, the L metric on %? is known as the Weil-Petersson metric. One has the following basic trichotomy for the structure of I?: (i) X > 0 => % = {pt}, consisting of the unique constant curvature metric of volume 1 on M = S. (ü) X = 0 => g = SL{2,Z)\SL(2,R)/SO(2) is the space of flat metrics on the torus. The Weil-Petersson metric is the complete, bi-invariant metric of finite volume on I?. (iii) X < 0 => If is the moduli space of hyperbolic metrics on a surface S^ of genus g > 1, and is the quotient of an open ball B~ by the properly discontinuous action of the Teichmüller modular group Fg. %