[Paper Review] Dual mixed volumes and isosystolic inequalities

Juan Carlos Álvarez Paiva|ArXiv.org|Aug 30, 2004

Advanced Differential Geometry Research37 references3 citations

TL;DR

This paper extends dual mixed volumes to star bodies in cotangent bundles and uses this framework to establish isosystolic inequalities for Finsler metrics and Hamiltonian systems. By introducing relative invariants $\tilde{W}_k(M;L,L_0)$ and leveraging Holmes-Thompson volume and systolic invariants, it proves that $\frac{\mathrm{sys}_k(M,L)^n}{\mathrm{sys}_k(M,L_0)^n} \leq \tilde{W}_k(M;L,L_0)^n \leq \frac{\mathrm{vol}(M,L)^{n-k}}{\mathrm{vol}(M,L_0)^{n-k}}$, generalizing Pu’s and Berger’s theorems to Finsler geometry and Hamiltonian dynamics.

ABSTRACT

The theory of dual mixed volumes is extended to star bodies in cotangent bundles and is used to prove several isosystolic inequalities for Hamiltonian systems and Finsler metrics.

Motivation & Objective

• To generalize Pu’s sharp isosystolic inequality for Riemannian metrics on $\mathbb{R}P^n$ to Finsler metrics.
• To extend Berger’s infinitesimal isosystolic inequality to Finsler metrics using Hamiltonian systems.
• To develop a dual mixed volume theory for star bodies in cotangent bundles to analyze systolic and volume invariants.
• To establish a Finsler generalization of the uniformization conjecture for $\mathbb{R}P^2$ via periodic geodesic flows.
• To prove isosystolic inequalities under conformal and commuting Hamiltonian conditions using averaging techniques.

Proposed method

• Define dual mixed volumes $\tilde{W}_k(M;L,L_0)$ for pairs of Finsler metrics on a compact $n$-manifold $M$ using the theory of dual mixed volumes on cotangent bundles.
• Use Holmes-Thompson volume as the volume measure in Finsler geometry, which is natural from a Hamiltonian perspective.
• Establish the inequality $\tilde{W}_k(M;L,L_0)^n \leq \frac{\mathrm{vol}(M,L)^{n-k}}{\mathrm{vol}(M,L_0)^{n-k}}$, with equality iff $L$ is a constant multiple of $L_0$.
• Prove $\tilde{W}_k(M;L,L_0) \geq \frac{\mathrm{sys}_k(M,L)}{\mathrm{sys}_k(M,L_0)}$ under three sets of hypotheses: conformality to invariant metrics, periodic geodesic flow, and commuting flows.
• Apply Hamiltonian averaging techniques to construct a path $K_t$ of metrics that agrees to first order with $L_t$ at $t=0$ and satisfies the isosystolic inequality.
• Use the fact that if $H_0$ has periodic flow, then any $H$ homogeneous of degree one can be written as $E + \{H_0, F\}$ with $\{H_0, E\} = 0$, enabling perturbation control.

Experimental results

Research questions

• RQ1 Can Pu’s isosystolic inequality for $\mathbb{R}P^2$ be extended to Finsler metrics?
• RQ2 Under what conditions does the isosystolic inequality $\frac{\mathrm{sys}_1^n(M,L)}{\mathrm{vol}(M,L)} \leq \frac{\mathrm{sys}_1^n(M,L_0)}{\mathrm{vol}(M,L_0)}$ hold for Finsler metrics on $\mathbb{R}P^n$?
• RQ3 What is the role of periodic geodesic flows and commuting Hamiltonians in deriving isosystolic inequalities?
• RQ4 Can Hamiltonian averaging techniques be used to generalize Berger’s infinitesimal isosystolic inequality to Finsler metrics?
• RQ5 Is there a Finsler version of the uniformization theorem for $\mathbb{R}P^2$ such that every reversible Finsler metric is conformal to one with periodic geodesic flow?

Key findings

• The inequality $\frac{\mathrm{sys}_1^n(M,L)}{\mathrm{vol}(M,L)} \leq \frac{\mathrm{sys}_1^n(M,L_0)}{\mathrm{vol}(M,L_0)}$ holds for Finsler metrics $L$ conformal to an invariant metric $L_0$ on $\mathbb{R}P^n$, generalizing Pu’s theorem.
• For $\mathbb{R}P^n$, if $L_0$ has periodic geodesic flow and $L$ commutes with $L_0$ in the Hamiltonian sense, then $\frac{\mathrm{sys}_1^n(M,L)}{\mathrm{vol}(M,L)} \leq \frac{\mathrm{sys}_1^n(M,L_0)}{\mathrm{vol}(M,L_0)}$.
• The Finsler metric $L$ on $SO(3)$ conformal to a left-invariant metric satisfies $\frac{\mathrm{sys}_1^3(SO(3),L)}{\mathrm{vol}(SO(3),L)} \leq \pi$, with equality iff $L$ is bi-invariant.
• For any smooth path $L_t$ of Finsler metrics on $\mathbb{R}P^n$ with $L_0$ having periodic geodesic flow, there exists a path $K_t$ agreeing to first order with $L_t$ at $t=0$ such that $\frac{\mathrm{sys}_1^n({\mathbb{R}}P^n,K_t)}{\mathrm{vol}({\mathbb{R}}P^n,K_t)} \leq \frac{\mathrm{sys}_1^n({\mathbb{R}}P^n,L_0)}{\mathrm{vol}({\mathbb{R}}P^n,L_0)}$, extending Berger’s result.
• The inequality $\tilde{W}_k(M;L,L_0)^n \leq \frac{\mathrm{vol}(M,L)^{n-k}}{\mathrm{vol}(M,L_0)^{n-k}}$ holds with equality iff $L$ is a constant multiple of $L_0$.
• The conjecture that every reversible Finsler metric on $\mathbb{R}P^2$ is conformal to one with periodic geodesic flow would imply the Finsler version of Pu’s theorem, as shown by Ivanov’s result.