Abstract. We associate to each stable Higgs pair (A0, Φ0) on a compact Riemann surface X a singular limiting configuration (A∞, Φ∞), assuming that det Φ has only simple zeroes. We then prove a desingularization theorem by constructing a family of solutions (At, tΦt) to Hitchin's equations which converge to this limiting configuration as t → ∞. This provides a new proof, via gluing methods, for elements in the ends of the Higgs bundle moduli space and identifies a dense open subset of the boundary of the compactification of this moduli space.

Abstract. Let M be a compact spin manifold. On the universal bundle of unit spinors we study a natural energy functional whose critical points, if dim M ≥ 3, are precisely the pairs (g, ϕ) consisting of a Ricci-flat Riemannian metric g together with a parallel g-spinor ϕ. We investigate the basic properties of this functional and study its negative gradient flow, the so-called spinor flow. In particular, we prove short-time existence and uniqueness for this flow.

Given a closed orientable Euclidean cone 3-manifold C with cone angles Ä and which is not almost product, we describe the space of constant curvature cone structures on C with cone angles < . We establish a regeneration result for such Euclidean cone manifolds into spherical or hyperbolic ones and we also deduce global rigidity for Euclidean cone structures. 57M50

We study the asymptotics of the natural L 2 metric on the Hitchin moduli space with group G = SU(2). Our main result, which addresses a detailed conjectural picture made by Gaiotto, Neitzke and Moore [GMN], is that on the regular part of the Hitchin system, this metric is well-approximated by the semiflat metric from [GMN]. We prove that the asymptotic rate of convergence for gauged tangent vectors to the moduli space has a precise polynomial expansion, and hence that the difference between the two sets of metric coefficients in a certain natural coordinate system also has polynomial decay. New work by Dumas and Neitzke shows that the convergence is actually exponential in directions tangent to the Hitchin section.

We investigate the local deformation space of 3-dimensional conemanifold structures of constant curvature κ ∈ {−1, 0, 1} and coneangles ≤ π. Under this assumption on the cone-angles the singular locus will be a trivalent graph. In the hyperbolic and the spherical case our main result is a vanishing theorem for the first L 2 -cohomology group of the smooth part of the cone-manifold with coefficients in the flat bundle of infinitesimal isometries. We conclude local rigidity from this. In the Euclidean case we prove that the first L 2 -cohomology group of the smooth part with coefficients in the flat tangent bundle is represented by parallel forms.