Collapsing Flat ${\rm{SU}}(2)$-Bundles to Spherical 3-Manifolds

Abstract

We present a geometric mechanism for the emergence of spherical $3$-manifolds from the superspace of Riemannian metrics associated with flat ${\rm{SU}}(2)$-bundles over closed orientable hyperbolic surfaces. Our main result shows that any homogeneous spherical 3-manifold $(S,g_{S})$ can be realized as a boundary point in the Gromov-Hausdorff closure of a superspace $\mathcal{S}(P)$, where $P$ is a flat ${\rm{SU}}(2)$-bundle over a closed orientable hyperbolic surface $(\Sigma,h_{\Sigma})$. We show that the convergence of the sequence of metric spaces towards the spherical limit is controlled by the order of the fundamental group of $S$ and the metric invariant of the hyperbolic base provided by the ratio between its area and its systole. In this framework, the problem of obtaining the sharpest upper bound error reduces to the classical problem of maximizing the systole function over the moduli space of hyperbolic Riemann surfaces. As a byproduct, we observe that certain arithmetic surfaces provide the best possible error estimates within this family. To illustrate these results, we show that, according to our mechanism, the Bolza surface yields the optimal error bound for the convergence toward the Poincar\'e homology sphere.