Limit law for Brownian cover time of the two dimensional torus

Let ${\mathcal{C}}_{\epsilon }$ denote the time it takes a Brownian motion to come within distance $\epsilon$ of every point in the two dimensional torus ${\mathbb{T}}^2$ of unit area. We prove that for an explicit non-random centering $m_{\epsilon }$, the variables $\{\sqrt{{\mathcal{C}}_{\epsilon }}-m_{\epsilon }\}$ converge in distribution as $\epsilon \to 0$, to a randomly shifted Gumbel law.