A Note on Finsler Version of Calabi-Yau Theorem

Songting Yin,Pan Zhang,Ruixin Wang

doi:10.1155/2018/1761608

Abstract

We generalize Calabi-Yau’s linear volume growth theorem to Finsler manifold with the weighted Ricci curvature bounded below by a negative function and show that such a manifold must have infinite volume.

Highlights

A Finsler space (M, F, dμ) is a differential manifold equipped with a Finsler metric F and a volume form dμ
A theorem due to Calabi and Yau states that the volume of any complete noncompact Riemannian manifold with nonnegative Ricci curvature has at least linear growth
The result was generalized to Riemannian manifolds with lower bound Ric ≥ −C/r(x)2 for some constant C, where r(x) is the distance function from some fixed point p

Summary

Introduction

A Finsler space (M, F, dμ) is a differential manifold equipped with a Finsler metric F and a volume form dμ. A theorem due to Calabi and Yau states that the volume of any complete noncompact Riemannian manifold with nonnegative Ricci curvature has at least linear growth (see [6, 7]). The result was generalized to Riemannian manifolds with lower bound Ric ≥ −C/r(x)2 for some constant C, where r(x) is the distance function from some fixed point p (see [8, 9]). As to the Finsler case, if the (weighted) Ricci curvature is nonnegative, the Calabi-Yau type linear volume growth theorem was obtained in [4, 10].

Results

Conclusion