Tightness for the Cover Time of the two dimensional sphere

Let $C^*_{ε,S^2}$ denote the cover time of the two dimensional sphere by a Wiener sausage of radius $ε$. We prove that $\sqrt{C^{*}_{ε,S^2} } -\sqrt{\frac{2A_{S^2}}π}(\log ε^{-1}-\frac14\log\log ε^{-1})$ is tight, where $A_{S^2}=4π$ denotes the Riemannian area of $S^2$.

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