[Paper Review] Volume, diameter and the minimal mass of a stationary 1-cycle

Alexander Nabutovsky, Regina Rotman|ArXiv.org|Jan 28, 2002

Geometric Analysis and Curvature Flows9 references4 citations

TL;DR

This paper establishes explicit upper bounds on the minimal mass of a non-trivial stationary 1-cycle on any closed Riemannian n-manifold, using geometric measure theory and minimax arguments. It proves that the minimal mass is at most $\frac{(n+2)!d}{3}$ in terms of diameter $d$, and $2(n+2)!\text{FillRad}(M^n)$ in terms of filling radius, with a volume-based upper bound involving $\text{vol}(M^n)^{1/n}$. These results provide curvature-free estimates for the length of shortest closed geodesics and related 1-cycles.

ABSTRACT

In this paper we present upper bounds on the minimal mass of a non-trivial stationary 1-cycle. The results that we obtain are valid for all closed Riemannian manifolds. The first result is that the minimal mass of a stationary 1-cycle on a closed n-dimensional Riemannian manifold M^n is bounded from above by (n+2)!d/3, where d is the diameter of a manifold M^n. The second result is that the minimal mass of a stationary 1-cycle on a closed Riemannian manifold M^n is bounded from above by 2(n+2)!Fill Rad(M^n) and, as a corollary, by 2(n+2)!(n+1)n^n(n!)^{1/2}(vol(M^n))^{1/n}, where Fill Rad(M^n) is the filling radius of the manifold, and vol(M^n) is its volume.

Motivation & Objective

• To establish curvature-free upper bounds on the minimal mass of a non-trivial stationary 1-cycle on any closed Riemannian n-manifold.
• To address Gromov’s question on whether the length of the shortest closed geodesic is bounded by a multiple of the diameter or volume root.
• To extend the Almgren-Pitts minimax method to parametrized 1-cycles and use homotopy-theoretic lifting techniques to construct such cycles.
• To provide explicit quantitative estimates on the mass and number of segments in optimal stationary 1-cycles.

Proposed method

• Constructs optimal 1-cycles as finite unions of geodesic segments with even multiplicity at vertices, ensuring the stationarity condition via vector sum cancellation at each point.
• Applies a minimax argument over maps from spheres into spaces of 1-cycles, inspired by Almgren and Pitts, to produce stationary varifolds.
• Uses a lifting technique to homotopically contract boundary cycles in the space of parametrized 1-cycles, relying on the ability to contract paired segments to midpoints.
• Leverages Gromov’s filling radius bounds in terms of volume to derive a volume-based upper bound for the minimal mass.
• Employs homotopy-theoretic obstruction theory via the Almgren isomorphism between homotopy groups of cycle spaces and homology groups of the manifold.
• Replaces full geometric measure theory machinery with elementary constructions on spaces $\Gamma_k$ of parametrized 1-cycles, while preserving the core geometric intuition.

Experimental results

Research questions

• RQ1 Can the minimal mass of a non-trivial stationary 1-cycle on a closed Riemannian n-manifold be bounded above without curvature assumptions?
• RQ2 Is there a universal constant $\tilde{c}(n)$ such that the minimal mass of a stationary 1-cycle is at most $\tilde{c}(n) \cdot \text{diam}(M^n)$?
• RQ3 Can the filling radius of a manifold be used to bound the minimal mass of a stationary 1-cycle, and how does this relate to volume?
• RQ4 Does the existence of such a bound imply improved estimates for the length of the shortest closed geodesic?
• RQ5 Can the minimax method in geometric measure theory be adapted to yield explicit, computable upper bounds on the mass of stationary 1-cycles?

Key findings

• The minimal mass of a non-trivial stationary 1-cycle on a closed n-dimensional Riemannian manifold $M^n$ is at most $\frac{(n+2)! \cdot d}{3}$, where $d$ is the diameter of $M^n$.
• An upper bound of $2(n+2)! \cdot \text{FillRad}(M^n)$ is established, and since $\text{FillRad}(M^n) \leq 2(n+1)n^n\sqrt{n!} \cdot \text{vol}(M^n)^{1/n}$, this yields a volume-based bound.
• The total number of geodesic segments (counted with multiplicity) in an optimal stationary 1-cycle of mass $\leq \tilde{c}(n)d$ is bounded in terms of $n$ alone.
• The construction produces a stationary 1-cycle of a special type—optimal and parametrized—whose mass satisfies the stated bounds.
• When $M^n$ is diffeomorphic to $S^2$, the method yields a closed geodesic, improving prior estimates on the shortest closed geodesic length.
• The results provide curvature-free upper bounds on the length of the shortest closed geodesic, extending earlier work by Croke, Maeda, and Sabourau.