Closed geodesics on hyperbolic surfaces with few intersections

TL;DR

The paper proves a sharp universal lower bound for the length of nonsimple closed geodesics with at least two self-intersections on oriented finite-type hyperbolic surfaces: $\ell(\Gamma) \ge 2\log(5+2\sqrt{6})$. The strategy combines a generalized collar lemma for asymmetric half-collars, a precise classification of short geodesics on pairs of pants, and winding-number analysis in collar regions to constrain geodesic length; these tools collectively show that any such geodesic must either lie in a pair of pants or exceed the bound. Equality is attained by a corkscrew geodesic on the thrice-punctured sphere, establishing $M_2=2\log(5+2\sqrt{6})$ as the exact value. The work advances understanding of minimal nonsimple geodesic lengths and provides a framework for exploring similar questions for larger self-intersection numbers on finite-type hyperbolic surfaces.

Abstract

We prove that, if a closed geodesic $Γ$ on a complete finite type hyperbolic surface has at least 2 self-intersections, then the length of $Γ$ has an lower bound $2\log(5+2\sqrt6)$, and the lower bound is sharp, attained on a corkscrew geodesic on a thrice punctured sphere.

Figures (5)

• Figure 1: The generalized collar $N_1(c)$
• Figure 2: A hexagon of $\Sigma_0$
• Figure 3: A corkscrew geodesic of $k=2$
• Figure 4: A covering of the annulus.
• Figure 5: A covering of $N_0(c_i)$

Theorems & Definitions (23)

• Definition 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2: Generalized collar lemma
• Lemma 2.3
• proof
• proof
• Lemma 2.4: Adams, A2817
• Remark 2.5
• Definition 3.1: Corkscrew geodesic, Y2801