We look for an $S^{1}$ invariant function $\phi$ on $\mathcal{M}$ satisfying the equation

(4.140)

$$T\pi^{*}\omega_{D}+dd^{c}\phi=T^{\frac{n-2}{n}}\omega$$

We write

(4.141)

$$d\phi=d_{D}\phi+\phi_{z}dz$$

where as before $d_{D}\phi$ is the differential along $D$ direction and $\phi_{z}=\partial_{z}\phi$ is the derivative along $z$ direction. Then

(4.142)

$$d^{c}\phi=d^{c}_{D}\phi+\phi_{z}h^{-1}\Theta,$$

and

(4.143)

$$dd^{c}\phi=d_{D}d_{D}^{c}\phi+dz\wedge(d^{c}_{D}\phi_{z})+d(\phi_{z}h^{-1})\wedge\Theta+\phi_{z}h^{-1}(\partial_{z}\tilde{\omega}-dz\wedge d_{D}^{c}h)$$

Since

(4.144)

$$T^{\frac{n-2}{n}}\omega=\pi^{*}\tilde{\omega}+dz\wedge\Theta,$$

we see (4.140) is equivalent to the system of equations

(4.145)

$$\begin{cases}\tilde{\omega}=T\omega_{D}+d_{D}d^{c}_{D}\phi+\phi_{z}h^{-1}\partial_{z}\tilde{\omega}\\ d_{D}^{c}\phi_{z}-\phi_{z}h^{-1}d_{D}^{c}h=0\\ d(\phi_{z}h^{-1})=dz.\end{cases}$$

To solve these (apparently overdetermined) equations, we first notice that the last equation in (4.145) is equivalent to

(4.146)

$$\phi_{z}h^{-1}=z+C$$

for a constant $C$. So we obtain ²² 2 In the case when $n=2$ for the classical Gibbons-Hawking ansatz this formula was derived by the authors together with Hans-Joachim Hein in the office of the first author at Stony Brook in the Fall of 2017.

(4.147)

$$\phi(z)=\int_{z_{0}}^{z}(u+C)hdu+\phi(z_{0})$$

for a function $\phi(z_{0})$ on $D$.

The second equation of (4.145) then holds automatically, and the first equation also follows after taking $\partial_{z}$. So in order for $\phi$ defined in (4.147) to satisfy (4.145), it suffices that at a fixed $z=T_{+}$ the following holds

(4.148)

$$T\omega_{D}+d_{D}d^{c}_{D}\phi=\tilde{\omega}-(T_{+}+C)\partial_{z}\tilde{\omega}$$

Comparing the cohomology class of both sides yields that $C$ must be zero. Then we can solve $\phi(T_{+})$ uniquely up to addition of a constant. After fixing a choice of $\phi(T_{+})$ we may define $\phi$ by

(4.149)

$$\phi(z)=\int_{T_{+}}^{z}uhdu+\phi(T_{+})$$

and we can view it as either a function on $Q_{T}$ or an $S^{1}$ invariant function on $\mathcal{M}$.

Notice from the above discussion we know for each fixed $z$, $\phi(z)$ is uniquely determined up to a constant on $D$ by the equation

(4.162)

$$T\omega_{D}+d_{D}d^{c}_{D}\phi(z)=\tilde{\omega}(z)-z\partial_{z}\tilde{\omega}(z),$$

and the integration formula (4.149) exactly gives a coherent way of fixing all the constants for each $z$, so the overall freedom in only up to a global constant. ³³ 3 maybe more geometric explanation if we have time

Notice by (3.349) we have for $z\gg 1$,

(4.163)

$$\tilde{\omega}(z)-z\partial_{z}\tilde{\omega}(z)-T\omega_{D}=\psi(z)-z\partial_{z}\psi(z)=\epsilon(z)$$

Standard elliptic estimate allows us to find a solution $\phi(T_{+})$ which is $\epsilon_{T}$. By (4.16) we obtain that for $z\geq C$

(4.164)

$$\phi(z)=C_{+}+T^{2-n}k_{+}^{-2}[\frac{(k_{+}z+T)^{n+1}}{n+1}-\frac{T(k_{+}z+T)^{n}}{n}]$$

where

(4.165)

$$C_{+}=\epsilon_{T}+\epsilon(z)-T^{2-n}k_{+}^{-2}[\frac{1}{n+1}T^{\frac{(n+1)(n-2)}{n}}-\frac{1}{n}T^{n-2}]$$

For the other end $z\leq-C$, similarly we have

(4.166)

$$\phi(z)-\phi(T_{-})=C_{-}+T^{2-n}k_{-}^{-2}[\frac{(k_{-}z+T)^{n+1}}{n+1}-\frac{T(k_{-}z+T)^{n}}{n}]$$

where

(4.167)

$$C_{-}=\epsilon_{T}+\epsilon(z)+T^{2-n}k_{-}^{-2}[\frac{1}{n+1}T^{\frac{(n+1)(n-2)}{n}}-\frac{1}{n}T^{n-2}]$$

To understand $\phi(T_{-})$ we need the following

Now we investigate (4.166).

(4.174)

$$\phi(z)-\phi(T_{-})=C_{-}+T^{2-n}k_{-}^{-2}[\frac{(k_{-}z+T)^{n+1}}{n+1}-\frac{T(k_{-}z+T)^{n}}{n}]+\epsilon_{T}+O(T^{-1})$$

We first notice that by (4.120)

(4.175)

$$-T^{3-n}k_{-}^{-2}\frac{(k_{-}z+T)^{n}}{n}=\frac{T}{k_{-}}(\log r_{-}-A_{-}+\epsilon_{T}+\epsilon(z))+\epsilon_{T}$$

We may also write by definition

(4.176)

$$\omega_{D}=-\frac{1}{k_{-}}dd^{c}\log r_{-}$$

So when $z\leq-C$, we have

(4.177)

$$T\pi^{*}\omega_{D}+dd^{c}\phi=T^{\frac{n-2}{n}}dd^{c}\phi_{-},$$

with

(4.178)

$$\phi_{-}\equiv\frac{1}{n+1}n^{\frac{n+1}{n}}k_{-}^{-\frac{n-1}{n}}(A_{-}+\epsilon_{T}+\epsilon(z)-\log r_{-})^{\frac{n+1}{n}}-T^{\frac{2}{n}}k_{-}^{-1}A_{-}+\epsilon_{T}+T^{\frac{2}{n}}\epsilon(z)$$

Similarly for $z\geq C$, we have

(4.179)

$$T\pi^{*}\omega_{D}+dd^{c}\phi=dd^{c}\phi_{+},$$

with

(4.180)

$$\phi_{+}\equiv\frac{1}{n+1}n^{\frac{n+1}{n}}(-k_{+})^{-\frac{n-1}{n}}(A_{+}+\epsilon_{T}+\epsilon(z)-\log r_{+})^{\frac{n+1}{n}}-T^{\frac{2}{n}}k_{+}^{-1}A_{+}+\epsilon_{T}+T^{\frac{2}{n}}\epsilon(z).$$