Let $\Sigma$ be a smooth open disk munished with a Riemannian metric $g$. We assume that $\Sigma$ has finite volume. Let $x$ be any point on $\Sigma$.
For $\epsilon>0$, let $m_\epsilon$ be the minimal length of a curve $\gamma$ that winds once around $x$ and stays at distance at most $\epsilon$ from the boundary (when $\epsilon$ is small, winding around $x$ is the same as "winding along the boundary").
question: Does there exists a constant $C_g$ such that $m_\epsilon\leq \frac{C_g}{\epsilon}$? If not, is there still a good bound on the divergence rate of $m_\epsilon$ as $\epsilon\to 0$?
remark: I know that $m_\epsilon$ is finite. Indeed, if $T$ is a tubular end of $\Sigma$, we double it across its boundary (including the boundary $\partial \Sigma$ "at infinity") to obtain a torus. We can then apply a refined version of the Loewner's torus inequality, to show that there is two curves that form a basis in homology and such that the product of there lengths is less than a constant times the volume. Back on $T$, it implies that there is a finite length curve between the two boundaries , and another one that winds along the boundary and stay at finite distance from the boundary. We can then go from "stay at finite distance" to "stay at small distance".
