Distinguishing Curve Types and Designer Metrics

TL;DR

This paper studies the inf invariant of filling curves on surfaces, exploring its properties, associated optimal metrics, and the resulting inf spectrum, revealing infinitely many pairs of curves with distinct invariants but identical self-intersection numbers.

Contribution

It introduces the concept of the inf invariant and optimal metrics for filling curves, constructing a family of curves with specific properties, and analyzing the inf spectrum on the moduli space.

Findings

Existence of infinitely many pairs of filling curves with distinct inf invariants but same self-intersection.

Construction of a two-parameter family of curves with coarse length bounds.

Establishment of coarse bounds for the inf spectrum on the moduli space.

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.