From the above Remark we know $d\Theta_{1}-\Upsilon$ is cohomologous to zero. The existence of a solution $\theta$ to $d(\Theta_{1}+\theta)=\Upsilon$ is obtained by adding the gauge fixing condition $d^{*}\theta=0$, and solving the elliptic system with Neumann boundary condition

(4.62)

$$\begin{cases}d\theta=\Upsilon-d\Theta_{1},\\ d^{*}\theta=0,\\ \theta(\nu)=0,\ \ \text{on}\ \ \partial\mathcal{V}.\end{cases}$$

on a tubular neighborhood $\mathcal{V}$ of $\mathcal{P}$ in $\mathbb{L}$. See Proposition 3.7 in [DS14] for example. By Proposition 4.5 we know $\Upsilon-d\Theta_{1}=O^{\prime}(s)$, particularly, $\Upsilon-d\Theta_{1}\in C^{\alpha}$ for all $\alpha\in(0,1)$. Hence standard elliptic regularity guarantees a solution $\theta\in C^{1,\alpha}$ and is smooth away from $\mathcal{P}$. Since both $\Upsilon$ and $\Theta_{1}$ are $S^{1}$-invariant, by averaging we may assume $\theta$ is $S^{1}$-invariant too, hence $\mathcal{L}_{\partial_{t}}\theta=0$ on the smooth part. Also since $\Upsilon$ and $d\Theta_{1}$ are pulled-back from the base $Q_{T}\setminus P$, we have

(4.63)

$$\partial_{t}\lrcorner\Upsilon=\partial_{t}\lrcorner d\Theta_{1}=0.$$

So we get

(4.64)

$$d(\partial_{t}\lrcorner\theta)=\mathcal{L}_{\partial_{t}}\theta-\partial_{t}\lrcorner(d\theta)=0.$$

This implies $\partial_{t}\lrcorner\theta$ is a constant. Now as we approach $\mathcal{P}$, the norm of $\partial_{t}$, with respect to the fixed metric on $\mathbb{L}$, must go to zero, hence we see

(4.65)

$$\partial_{t}\lrcorner\theta=0.$$

The higher regularity of $\theta$ follows just as in the proof of Lemma 3.22 in Section 3. ∎