The hyperkähler metric on the almost-Fuchsian moduli space

Abstract

Donaldson \[11] constructed a hyperkähler moduli space $\mathcal{M}$ associated to a closed oriented surface $\Sigma$ with genus$(\Sigma) \geq 2$. This embeds naturally into the cotangent bundle $T^\*\mathcal T(\Sigma)$ of Teichmüller space or can be identified with the almost-Fuchsian moduli space associated to $\Sigma$. The latter is the moduli space of quasi-Fuchsian threefolds which contain a unique incompressible minimal surface with principal curvatures in $(-1,1)$. Donaldson outlined various remarkable properties of this moduli space for which we provide complete proofs in this paper: On the cotangent-bundle of Teichmüller space, the hyperkähler structure on $\mathcal M$ can be viewed as the Feix–Kaledin hyperkähler extension of the Weil–Petersson metric. The almost-Fuchsian moduli space embeds into the SL$(2,\mathbb C)$-representation variety of $\Sigma$ and the hyperkähler structure on $\mathcal M$ extends the Goldman holomorphic symplectic structure. Here, the natural complex structure corresponds to the second complex structure in the first picture. Moreover, the area of the minimal surface in an almost-Fuchsian manifold provides a kähler potential for the hyperkähler metric. The various identifications are obtained using the work of Uhlenbeck \[33] on germs of hyperbolic 3-manifolds, an explicit map from $\mathcal M$ to $\mathcal T(\Sigma)\times \overline{\mathcal T(\Sigma)}$ found by Hodge \[20], the simultaneous uniformization theorem of Bers \[2], and the theory of Higgs bundles introduced by Hitchin \[18].