4.8. Harmonic analysis I: periodic Euclidean region [046I]

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4.8. Harmonic analysis I: periodic Euclidean region

The refined mapping properties of the Euclidean Green operator $\Delta_{a}^{-1}$ on $(\mathbb{C}^{*})^{2}_{\eta_{1},\eta_{2}}\times\mathbb{R}_{\mu}$ follow Section 3.5 almost verbatim:

Proposition 4.31.

(Periodic Euclidean region) Let $-3<\delta<0$ . Let $f$ be a function compactly supported in $\mathcal{B}^{-}_{\nu}\cap\{R>A^{-1/2}\}$ with $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,0}}\leq 1$ (respectively $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta}}\leq 1$ ). Then $\Delta_{a}^{-1}f$ satisfies the $g_{a}$ -Hessian bound on $\mathcal{B}^{-}_{\nu}\cap\{R\gtrsim A^{-1/2}\}$ ,

\left\lVert\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f\right\rVert_{C^{k,\alpha}_{\delta,0}}\leq C\nu,\quad\text{resp. }\left\lVert\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f\right\rVert_{C^{k,\alpha}_{\delta}}\leq C.

Remark 4.8.

The regularity of $\Delta_{a}^{-1}f$ in $\{R\lesssim A^{-1/2}\}$ is well controlled by $\Delta_{a}$ -harmonicity.

This allows us to correct the volume form error sufficiently away from $S$ as in proposition 3.23. From now on $1\ll\nu\ll A^{3/8}$ .

Proposition 4.32.

Let $1\ll\nu\ll A^{3/8}$ . Then there is a real valued function $\varphi_{1}$ on $\mathcal{B}^{-}_{\nu}$ , solving the generalised Gibbons-Hawking equation on $\mathcal{B}^{-}_{\nu}\cap\{\tilde{\ell}>2\}$

V_{(2)}=V_{(1)}+\frac{\partial^{2}\varphi_{1}}{\partial\mu\partial\mu},\quad W^{p\bar{q}}_{(2)}=W^{p\bar{q}}_{(1)}-4\frac{\partial^{2}\varphi_{1}}{\partial\eta_{p}\partial\bar{\eta}_{q}},\quad\det(W^{p\bar{q}}_{(2)})=V_{(2)}.

Morever $\varphi_{1}$ is $\Delta_{a}$ -harmonic on $\mathcal{B}^{-}_{\nu}\cap\{\tilde{\ell}<1\}$ , and

\left\lVert\nabla^{2}_{g_{a}}\varphi_{1}\right\rVert_{C^{k,\alpha}_{-1,0}(\mathcal{B}^{-}_{\nu}\cap\{\tilde{\ell}\gtrsim 1\})}\leq C\nu^{3}A^{-3/4},

and $|\nabla^{2}_{g_{a}}\varphi_{1}|_{g_{a}}\leq C\nu^{3}A^{-3/2}$ on $\mathcal{B}^{-}_{\nu}$ . In particular the matrix $(W^{p\bar{q}}_{(2)})$ is positive definite and $V_{(2)}$ is positive on $\mathcal{B}^{-}_{\nu}$ .

We obtain by the generalised Gibbons-Hawking construction $(g^{(2)},\omega^{(2)},J,\Omega)$ associated to the data $V_{(2)}$ and $W^{p\bar{q}}_{(2)}$ , and identify its ambient space as $M^{-}_{\nu}$ . The new $S^{1}$ -connection $\vartheta^{(2)}$ is related to $\vartheta$ by

\vartheta^{(2)}-\vartheta=\sqrt{-1}(\frac{\partial^{2}\varphi^{-}}{\partial\eta_{p}\partial\mu}d\eta_{p}-\frac{\partial^{2}\varphi^{-}}{\partial\bar{\eta}_{p}\partial\mu}d\bar{\eta}_{p}).

The new volume form error $E^{(2)}$ is supported in $\{\ell\lesssim 1\}$ with bound

(4.31)

E^{(2)}=\frac{\det(W^{p\bar{q}}_{(2)})}{V_{(2)}}-1,\quad\left\lVert E^{(2)}\right\rVert_{C^{1,\alpha}_{-1}(M^{-}_{\nu})}\leq CA^{-3/4}\nu^{2},

and in particular $\left\lVert E^{(2)}\right\rVert_{C^{\alpha}_{-1-\epsilon}(M^{-}_{\nu})}\leq CA^{-3/4}\nu^{2}$ .

Henceforth the holomorphic structures will be fixed, and can be identified building on results in Section 4.6. The new holomorphic differentials $d\log Z_{3},d\log Z_{4}$ are related to $d\log z_{3},d\log z_{4}$ by

\begin{cases}d\log Z_{3}=d\log z_{3}+d(\frac{\partial\varphi_{1}}{\partial\mu}),\\ d\log Z_{4}=d\log z_{4}-d(\frac{\partial\varphi_{1}}{\partial\mu}),\end{cases}

whence we find holomorphic coordinates by integration

(4.32)

Z_{3}=z_{3}\exp(\frac{\partial\varphi_{1}}{\partial\mu}),\quad Z_{4}=z_{4}\exp(-\frac{\partial\varphi_{1}}{\partial\mu}),

which satisfy the functional equation

Z_{3}Z_{4}=z_{3}z_{4}=1-z_{1}-z_{2}=1-e^{2\pi i\eta_{1}}-e^{2\pi i\eta_{2}}.

Following Lemma 4.28 and Proposition 4.29,

Proposition 4.33.

(Holomorphic structure) The map

M^{-}_{\nu}\to\{Z_{3}Z_{4}=1-z_{1}-z_{2}\}\subset\mathbb{C}^{2}_{Z_{3},Z_{4}}\times(\mathbb{C}^{*})^{2}_{z_{1},z_{2}}

extends continuously over the singular locus $S\cap M^{-}_{\nu}$ and defines a holomorphic open embedding under the complex structure $J$ . The $S^{1}$ -action is identified as

e^{i\theta}\cdot(z_{1},z_{2},Z_{3},Z_{4})=(z_{1},z_{2},e^{i\theta}Z_{3},e^{-i\theta}Z_{4}),

and the holomorphic volume form is $\Omega=-\frac{\sqrt{-1}}{4\pi^{2}z_{1}z_{2}}dz_{2}\wedge dZ_{3}\wedge dZ_{4}.$ The Kähler structure is $C^{1,\alpha}$ -regular near $S$ .

Proposition 4.34.

(Symplectic structure) The integral $\int_{T^{2}}\omega^{(2)}=\text{Im}(a_{2\bar{1}}).$

Proof.

The new Kähler form $\omega^{(2)}$ is cohomologous to $\omega^{(1)}$ by the formula

\omega_{-}=\omega^{(1)}+d(\sqrt{-1}\frac{\partial\varphi^{-}}{\partial\eta_{p}}d\eta_{p}-\sqrt{-1}\frac{\partial\varphi^{-}}{\partial\bar{\eta}_{p}}d\bar{\eta}_{p}).

The claim then follows from Lemma 4.14. ∎