Quantitative estimate of diameter for weighted manifolds under integral curvature bounds and $\varepsilon$-range

TL;DR

This paper extends compactness theorems to weighted manifolds with integral curvature bounds, utilizing the $ ext{ extgreek extbeta}$-range and segment inequality, to estimate diameters under various effective dimension conditions.

Contribution

It generalizes existing compactness results to weighted manifolds with lower bounds on weighted Ricci curvature, incorporating the $ ext{ extgreek extbeta}$-range and segment inequality extensions.

Findings

Extended compactness theorems to weighted manifolds.

Provided diameter estimates under integral curvature bounds.

Generalized segment inequality for weighted manifolds.

Abstract

In this article, we extend the compactness theorems proved by Sprouse and Hwang-Lee to a weighted manifold under the assumption that the weighted Ricci curvature is bounded below in terms of its weight function. With the help of the $\varepsilon$-range, we treat the case that the effective dimension is at most $1$ in addition to the case that the effective dimension is at least the dimension of the manifold. To show these theorems, we extend the segment inequality of Cheeger-Colding to a weighted manifold.