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.

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 review recent work on the compactification of the moduli space of Hitchin's self-duality equation. We study the degeneration behavior near the ends of this moduli space in a set of generic directions by showing how limiting configurations can be desingularized. Following ideas of Hitchin, we can relate the top boundary stratum of this space of limiting configurations to a Prym variety. A key rôle is played by the family of rotationally symmetric solutions to the self-duality equation on C, which we discuss in detail here.

We use the Yang-Mills gradient flow on the space of connections over a closed Riemann surface to construct a Morse chain complex. The chain groups are generated by Yang-Mills connections. The boundary operator is defined by counting the elements of appropriately defined moduli spaces of Yang-Mills gradient flow lines that converge asymptotically to Yang-Mills connections. RésuméOn utilise le flot du gradient de la fonctionelle de Yang-Mills définie sur l'espace des connexions d'une surface de Riemann fermée afin de construire un complexe de Morse. Les groupes de ce complexe sont engendrés par les connexions de Yang-Mills. La différentielle compte les éléments d'un espace de modules convenablement défini qui consiste des trajectoires du flot de gradient convergeant asymptotiquement vers des connexions de Yang-Mills.

Abstract. We prove a gluing theorem for solutions (A0, Φ0) of Hitchin's self-duality equations with logarithmic singularities on a rank-2 vector bundle over a noded Riemann surface Σ0 representing a boundary point of Teichmüller moduli space. We show that every nearby smooth Riemann surface Σ1 carries a smooth solution (A1, Φ1) of the self-duality equations, which may be viewed as a desingularization of (A0, Φ0).