4.11. Ooguri-Vafa type metrics on the negative vertex [046Y]

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4.11. Ooguri-Vafa type metrics on the negative vertex

We discuss the geometric aspects of the Ooguri-Vafa type metric $(g_{-},\omega_{-},\Omega)$ .

Combining deviation estimates in Proposition 4.13, 4.32 and Theorem 4.38,

Corollary 4.39.

(Exponential decay to semiflat metric away from $S$ ) In the situation of Theorem 4.38, for $\tilde{\ell}\gtrsim 1$ the deviation of $\omega_{-}$ from its zeroth Fourier mode $\overline{\omega}_{-}$ decays exponentially:

|\omega_{-}-\overline{\omega}_{-}|\leq CA^{-3/4}\nu e^{-\kappa\tilde{\ell}}.

The transverse structure along $S$ follows immediately from Proposition 4.19 and the metric deviation estimates.

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.

The smooth topology emerges a posteriori after solving the Monge-Ampère equation as a consequence of regularity theory.

Proposition 4.41.

The metric structure $(g_{-},\omega_{-},J,\Omega)$ extends smoothly to a Calabi-Yau metric on $M^{-}_{\nu}$ .

Proof.

The holomorphic coordinates extend across $S$ , so induces a smooth structure on $M^{-}_{\nu}$ . The metric $\omega_{-}$ has $C^{\alpha}$ -regularity and is Calabi-Yau, so standard regularity theory of complex Monge-Ampère equation implies that it is smooth. ∎