Corollary 4.40 . [0470]

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Corollary 4.40.

(Transverse Taub-NUT metrics) Under suitable gauge choices for the $S^{1}$ -connection, the metric $g_{-}$ restricted over the disc $\{\xi_{2}=0,R\lesssim A^{1/4}\}$ is approximated by the Taub-NUT metric:

|(g_{-}-g_{\text{NUT}})|_{\xi_{2}=0}|_{g_{\text{NUT}}}\leq CA^{-3/4}\nu.