Length of a closed geodesic in 3-manifolds of positive scalar curvature

TL;DR

The paper proves that every closed 3-manifold with scalar curvature bounded below by $R_g\ge6$ contains a non-trivial closed geodesic of length at most $22500$, confirming Gromov’s conjecture in dimension three. The authors reduce to the $3$-sphere case using the topological classification under positive scalar curvature, then construct a tree-foliation by $L$-short embedded $2$-spheres with controlled area and diameter via a discrete mean curvature flow with surgery. They obtain a two-parameter family of short curves by sweeping each sphere in the foliation, deform this family to a uniformly short sweepout, and apply Morse theory on the free loop space to extract a closed geodesic of Morse index at most $2$. The result reveals rigidity in the space of 3-manifolds with $R_g\ge6$ and provides a quantitative bound that, while not sharp, demonstrates a robust link between scalar curvature and geodesic length with potential implications for systolic phenomena in low dimensions.

Abstract

Let $M$ be a closed $3$-dimensional Riemannian manifold with positive scalar curvature, $R_g \geq 6$. We show that $M$ contains a non-trivial closed geodesic of length less than $22500$. This confirms a conjecture of M. Gromov in dimension $3$.

Figures (15)

• Figure 1: A 3-sphere with many splines and $R\geq 6$
• Figure 2: $L$-short $\Sigma_2$ after cut-and-paste surgeries inside $B$
• Figure 3: Case 2: $a_i$ is contained in a connected component $D$ of $S$
• Figure 4: $\varepsilon$-neck-pinching along a disk $D$
• Figure 5: $S \cap B_r(p)$ and $\Sigma \cap B_r(p)$ are isotopic.

Theorems & Definitions (41)

• Theorem A
• Theorem B
• Conjecture 1.1
• Theorem 2.1
• proof
• Definition 3.1: $L$-Shortness
• Remark 3.2
• Definition 3.3
• Theorem 3.4
• Theorem 3.5