Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $\Sigma$ be a closed, oriented, smooth surface. Denote by $\mathcal{M}^{1}(\Sigma)$ the deformation space of unit volume Riemannian metrics on $\Sigma:$ here we consider two metrics equivalent if they differ via an orientation preserving diffeomorphism which is homotopic to the identity. For the time being, let us ignore whether we choose a Sobolev topology on this space, or the more natural, but difficult, Frechet topology. Given a metric $g\in\mathcal{M}^{1}(\Sigma),$ we can identify the tangent space $T_{g}(\mathcal{M}^{1}(\Sigma))$ with the space of $g$-divergence free, symmetric two tensors on $\Sigma.$ Given two such tensors $\alpha, \beta \in T_{g}(\mathcal{M}^{1}(\Sigma)),$ the $L^2$-pairing is defined via the expression (in local coordinates), \begin{align} \langle \alpha, \beta \rangle_{g}:=\int_{\Sigma} \alpha^{ij} \beta_{ij}\ dV_{g}, \end{align} where the indices of $\alpha$ are raised using the metric $g.$ The Hodge star operator associated to $g$ yields an almost-complex structure on $T_{g}(\mathcal{M}^{1}(\Sigma))$ given by viewing a symmetric two tensor as a one form with values in the cotangent bundle $T^{*}(\Sigma),$ and applying the Hodge star operator to the one form part. The question is the following: ignoring the fact of topologies and that we're in infinite dimensions, does the $L^2$-metric with this almost complex structure make $\mathcal{M}^{1}(\Sigma)$ a Kahler manifold. In particular, is this almost-complex structure constant along geodesics for the $L^2$-metric.

Some history is in order, the space of isotopy classes of constant negative curvature metrics $\mathcal{F}(\Sigma)$ on $\Sigma$ is identified (via the uniformization theorem) with the space of istopopy classes of complex structures on $\Sigma,$ which is the classical Teichmuller space. Under this identification, it has been known for a long while (nicely exposed in the book of Tromba "Teichmuller theory in Riemannian Geometry) that the Weil-Petersson pairing of holomorphic quadratic differentials can be identified with the $L^2$-metric. This should not be surprising as the Weil-Petersson pairing is an $L^2$-pairing. It was shown, first by Ahlfors, that the Weil-Petersson metric is Kahler. This is evidence for the question I ask above.

Thanks in advance for any insight into this question!

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.