Distinguishing Curve Types and Designer Metrics

Abstract

Let $\gamma$ be a filling curve on a topological surface $\Sigma$ of genus $g \geq 2$. The inf invariant of $\gamma$, denoted $m_{\gamma}$, is the infimum of the length function on the space of marked hyperbolic structures on $\Sigma$. This infimum is realized at a unique hyperbolic structure, $X_{\gamma}$, which we call the optimal metric associated to $\gamma$. In this paper, we investigate properties of the inf invariant and its associated optimal metric. Starting from a filling curve and a separating curve, we construct a two integer parameter family of curves for which we derive coarse length bounds and qualitative properties of their associated optimal metrics. In particular, we show that there are infinitely many pairs of filling curves, each pair having distinct $\text{inf}$ invariants but the same self-intersection number. The inf invariants give rise to a natural spectrum, we call the inf spectrum, associated to the moduli space of the surface. We provide coarse bounds for this spectrum.