Baptista's adiabatic metric conjecture for Abelian vortices

Let $\Sigma$ be the underlying compact Riemann surface, let $\nu_{k,0}(s)$ be the relevant open subset of the vortex moduli space, and let

$$\Phi_s:Hol_r(\Sigma,\mathbb{CP}^{k-1})\to\nu_{k,0}(s)$$

be the diffeomorphic correspondence. Let $g_s$ denote the natural $L^2$ metric on $\nu_{k,0}(s)$, and write $g_s^*=\Phi_s^*g_s$ for its pullback. The ordinary $L^2$ metric on $Hol_r(\Sigma,\mathbb{CP}^{k-1})$ is denoted by $\langle\cdot,\cdot\rangle_{L^2}$. Baptista's conjecture. On $Hol_r(\Sigma,\mathbb{CP}^{k-1})\backsimeq\nu_{k,0}(s)$, the metrics $g_s^*$ converge smoothly, as $s\to\infty$, to a multiple of the ordinary $L^2$ metric $\langle\cdot,\cdot\rangle_{L^2}$. The paper proves the convergence of the functions determining the complex gauges, establishing this conjecture on vortex dynamics; the stated metric convergence is the adiabatic-limit description of the vortex moduli-space metric.