Comparison Theorems on Weighted Finsler Manifolds and Spacetimes with ϵ-Range

Shin-Ichi Ohta,Ettore Minguzzi,Yufeng Lu

doi:10.1515/agms-2020-0131

Abstract

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 ϵ-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.

Highlights

A weighted manifold is a pair given by a manifold, equipped with some metric, and a weight function on it
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 ε-range introduced in our previous paper, that provides a natural viewpoint of interpolating weighted Ricci curvature conditions of di erent e ective dimensions

Summary

Introduction

A weighted manifold is a pair given by a manifold, equipped with some metric, and a weight function on it. Gromov volume comparison theorem (in the latter the weight function ψ is induced from a given measure m on M), in both weighted Finsler manifolds and weighted Finsler spacetimes We remark that those results for ε ≠ (N − )/(N − n) with N < or for ε ≠ with N ∈ [n, +∞] are new even in the weighted Riemannian setting. For the Bonnet–Myers and Laplacian comparison theorems on Finsler manifolds, our results cover both the unweighted case [3] and the weighted case associated with measures [34, 39]; this uni cation is not included in the literature. Some arguments could be uni ed to a single framework, we shall discuss the Finsler and Lorentz–Finsler cases rather separately and present the proofs of comparison theorems in their each common languages, for the sake of accessibility and hopefully motivating interactions between Riemannian and Lorentzian geometries

Preliminaries for Finsler manifolds

By using

Comparison theorems on weighted Finsler manifolds

We also de ne

Laplacian of u by

Therefore we have

Finsler spacetimes

Therefore we de ne n

Comparison theorems on weighted Finsler spacetimes