Comparison theorems on weighted Finsler manifolds and spacetimes with $\epsilon$-range

TL;DR

This paper extends classical comparison theorems like Bonnet-Myers and Bishop-Gromov to weighted Finsler manifolds and spacetimes, using an innovative $\epsilon$-range approach that interpolates various curvature conditions.

Contribution

It introduces new comparison theorems for weighted Finsler geometries with an $\epsilon$-range, generalizing existing results and providing a unified framework.

Findings

Established Bonnet-Myers, Laplacian, and Bishop-Gromov theorems for weighted Finsler manifolds and spacetimes.

The $\epsilon$-range approach interpolates different curvature conditions, broadening applicability.

Some results are novel even for weighted Riemannian manifolds, extending prior comparison theorems.

Abstract

We establish the Bonnet-Myers theorem, Laplacian comparison theorem, and Bishop-Gromov volume comparison theorem for weighted Finsler manifolds as well as weighted Finsler spacetimes, of weighted Ricci curvature bounded below by using the weight function. These comparison theorems are formulated with $\epsilon$-range introduced in our previous paper, that provides a natural viewpoint of interpolating weighted Ricci curvature conditions of different effective dimensions. Some of our results are new even for weighted Riemannian manifolds and generalize comparison theorems of Wylie-Yeroshkin and Kuwae-Li.