The hyperkähler metric on the almost-Fuchsian moduli space

Abstract

Donaldon constructed a hyperk\"ahler moduli space $\mathcal{M}$ associated to a closed oriented surface $\Sigma$ with $\textrm{genus}(\Sigma) \geq 2$. This embeds naturally into the cotangent bundle $T^*\mathcal{T}(\Sigma)$ of Teichm\"uller space or can be identified with the almost-Fuchsian moduli space associated to $\Sigma$. The later 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 Teichm\"uller space, the hyperk\"ahler structure on $\mathcal{M}$ can be viewed as the Feix--Kaledin hyperk\"ahler extension of the Weil--Petersson metric. The almost-Fuchsian moduli space embeds into the $\textrm{SL}(2,\mathbb{C})$-representation variety of $\Sigma$ and the hyperk\"ahler 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 K\"ahler potential for the hyperk\"ahler metric. The various identifications are obtained using the work of Uhlenbeck on germs of hyperbolic $3$-manifolds, an explicit map from $\mathcal{M}$ to $\mathcal{T}(\Sigma)\times \bar{\mathcal{T}(\Sigma)}$ found by Hodge, the simultaneous uniformization theorem of Bers, and the theory of Higgs bundles introduced by Hitchin.