Short closed geodesics and the Willmore energy

TL;DR

The paper proves a universal lower bound for the length of the shortest closed geodesic on $\mathbb{S}^2$ immersed in $\mathbb{R}^n$ in terms of the Willmore energy and area, valid for $\mathcal{W}(f)<6\pi$: $\ell(\mathbb{S}^2,g_f) \ge C(n)(6\pi-\mathcal{W}(f))\sqrt{\mathcal{A}(f)}$. The argument combines a tiling of $\mathbb{S}^2$ by a (potentially self-intersecting) closed geodesic, Gauss–Bonnet on each tile, and divergence-based bounds (Michael–Simon type inequalities) to control curvature and area contributions, yielding a quantitative length bound. The energy threshold $6\pi$ is sharp, and the inequality does not extend to surfaces of positive genus; the paper also demonstrates the existence of short geodesics on thin surfaces of revolution and via sphere inversion with spherical replacement, illustrating sharpness and providing injectivity-radius consequences. Together, these results contribute a rigidity-type relation between Willmore energy and systolic-type invariants for two-dimensional immersions, linking conformal control, curvature, and the global geometry of the surface.

Abstract

We prove a lower bound on the length of closed geodesics for spheres with Willmore energy below $6π$. The energy threshold is optimal and the inequality cannot be extended to surfaces of higher genus. Moreover, we discuss consequences for the injectivity radius.

Figures (4)

• Figure 1: Short closed geodesics on $\Sigma_3$ and $\Sigma_5$.
• Figure 2: Removing a vertex from a connected component whose boundary has self-intersections.
• Figure 3: Construction of the cone $C$ with opening angle $\omega\in (\pi, 2\pi)$.
• Figure 4: Plots of closed geodesics with self-intersections on the spheroid for different parameters $b$, $c$, $\varepsilon$, $N$.

Theorems & Definitions (38)

• Theorem 1.1
• Corollary 1.2
• Example 1.3: Optimality of the $6\pi$-threshold
• Example 1.4: Optimality of zero genus
• Example 1.5: Small Willmore energy and self-intersections
• Example 1.7: Unbounded Gauss curvature
• Lemma 2.1
• proof
• Theorem 2.2
• proof