The maximum diam theorem on Finsler manifolds

TL;DR

This paper proves a maximum diameter theorem for Finsler manifolds with positive weighted Ricci curvature, characterizing when such spaces are isometric to a Finsler sphere, and relates this to the first eigenvalue of the Finsler-Laplacian.

Contribution

It establishes a Finsler analogue of the classical sphere theorem, linking diameter, curvature bounds, and spectral properties to characterize Finsler spheres.

Findings

Maximal diameter implies the manifold is a Finsler sphere.

First eigenvalue attains its lower bound only on Finsler spheres.

Explicit first eigenfunction on the Finsler sphere is derived.

Abstract

We prove that, for a Finsler space, if the weighted Ricci curvature is bounded below by a positive number and the diam attains its maximal value, then it is isometric to a standard Finsler sphere. As an application, we show that the first eigenvalue of the Finsler-Laplacian attains its lower bound if and only if the Finsler manifold is isometric to a standard Finsler sphere, and moreover, we obtain an explicit 1-st eigenfunction on the sphere.