arXiv2026
Let $B$ be a compact Riemann surface and $B_0\subseteq B$ a hyperbolic surface obtained by removing finitely many disjoint closed disks. Fix a nontrivial loop $α$ in $B_0$. For $s\ge0$, let $L(α,s)$ denote the supremum, over all finite subsets $S\subset B_0$ with $\#S\le s$, of the minimal Kobayashi length of a loop in $B_0\smallsetminus S$ that is freely homotopic to $α$ in $B_0$. Phung in [14] proved that $L(α,s)$ grows at most linearly and at least as $\sqrt{s}/\log s$. We sharpen the upper bound to $O(\sqrt{s+1})$, which determines $\lim_{s\to\infty}\log L(α,s)/\log s=1/2$, answering [14, Question 1.4]. As an application, we improve Phung's counting bound for generalized integral points on abelian varieties over complex function fields. For an abelian variety of dimension $n$ over $\mathbb C(B)$ satisfying his geometric hypotheses, Phung proved a bound of order $(s+1)^{2nk}$ modulo the constant trace, where $k$ is the minimal number of generators of $π_1(B_0)$. We explain his method in detail and insert our stronger length estimate to obtain $(s+1)^{nk}$, halving the exponent. Counting in the Mordell--Weil lattice of rank $r$ gives the more precise bound $O((s+1)^{r/2})$ and a linear bound in $s+1$ for the canonical heights, with the family and $B_0$ fixed. For normal-crossings $D$ with Cartier model closure, this proves the fixed-model geometric Lang-Vojta conjecture in the stated abelian setting, without an exceptional set.