Short Simple Geodesic Loops on a 2-Sphere

TL;DR

The paper addresses the problem of finding two short simple geodesic loops anchored at a fixed point $p$ on an analytic Riemannian 2-sphere of diameter $d$, providing explicit length bounds. It adapts the Lusternik--Schnirelmann framework to basepointed loops by developing a $p$-modified disk flow, constructs short meridional slicings via Berger's lemma, and extends shortening to multi-parameter families to realize nontrivial min--max cycles. The main result shows the existence of two distinct simple geodesic loops at $p$ with lengths at most $8d$ and $14d$, respectively, offering a basepoint-specific quantitative analogue of the classical closed-geodesic theorems. These techniques contribute to the understanding of the length spectrum on $S^2$ and demonstrate how basepoint geometry can yield explicit geometric-analytic bounds in two-dimensional Riemannian geometry.

Abstract

The classic Lusternik--Schnirelmann theorem states that there are three distinct simple periodic geodesics on any Riemannian 2-sphere $M$. It has been proven by Y. Liokumovich, A. Nabutovsky and R. Rotman that the shortest three such curves have lengths bounded in terms of the diameter $d$ of $M$. We show that at any point $p$ on $M$ there exist at least two distinct simple geodesic loops (geodesic segments that start and end at $p$) whose lengths are respectively bounded by $8d$ and $14d$.

Figures (10)

• Figure 1: An example of the modified disk flow in $B_1$. The dashed arc contains the point corresponding to $t=0$.
• Figure 2: Eliminating a digon in order to homotope an arc to the corresponding geodesic segment. The dashed line is $\eta_i^j$ and the solid line is an arc of $\gamma$.
• Figure 3: A minimizing geodesic $\eta$ that crosses the pair of rays $\alpha'$ and $\beta'$ must also cross the inner pair of rays $\alpha$ and $\beta$.
• Figure 4: Left: there exists a diameter that lies outside the sector given by the endpoints of $\gamma_0$ and intersects $\gamma'$ at least twice. Right: the curve obtained by replacing a section of $\gamma'$ with a portion of a diameter, which is shorter than $\gamma'$.
• Figure 5: Left: the sector bounded by $\gamma_i$ when the final loop of $\gamma_\infty$ is $\eta$. Right: the two sector bounded by $\gamma_i$ when the final loop of $\gamma_\infty$ is $-\eta$.

Theorems & Definitions (38)

• Theorem 1
• Definition 1: $p$-Admissible Curves
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof