Lemma 4.10 . [02HE]

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Lemma 4.10.

As $\epsilon\rightarrow 0$ the harmonic function $h_{\epsilon}$ converges to the constant function $1$ . The convergence is in $C^{k,\alpha}$ on the complement of the union of balls of radius $\epsilon^{\beta}$ around the punctures, where $\beta>0$ is any number such that $\beta^{-1}>k+1+\alpha$ . In particular, $M^{\textup{gh}}_{\epsilon}$ collapses to the flat orbifold $\mathbb{T}/\tau$ with bounded curvature away from the punctures.

Proof.

The first statement is a simple application of Lemma 4.7, since close to each puncture we have

\rho^{k}|\nabla^{k}(h_{\epsilon}-1)|\leq C\epsilon\rho^{-1},

where $\rho$ is the distance from the puncture. It follows that away from the punctures $g^{\textup{gh}}_{\epsilon}$ is $C^{k,\alpha}_{loc}$ –close to the $\mathbb{Z}_{2}$ quotient of $g_{\mathbb{T}}+\epsilon^{2}\theta^{2}$ for any $k\geq 0$ and $\epsilon$ sufficiently small. ∎