The moduli of flat PU(p, p)-structures with large Toledo invariants

For a compact Riemann surface X of genus g > 1, Hom(π₁ (X), PU(p, q))/PU(p, q) is the moduli space of flat PU(p, q)-connections on X. There are two invariants, the Chern class c and the Toledo invariant τ associated with each element in the moduli. The Toledo invariant is bounded in the range -2min(p, q)(g-1) ≤ τ ≤ 2min(p, q)(g - 1). This paper shows that the component, associated with a fixed τ > 2(max(p, q) - 1)(g - 1) (resp. τ < -2(max(p, q) - 1)(g - 1)) and a fixed Chern class c, is connected (The restriction on τ implies p = q).