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Quantitative estimate of diameter for weighted manifolds under integral curvature bounds and $\varepsilon$-range
[article]
2022
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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.
doi:10.48550/arxiv.2202.06680
fatcat:iafovxnrejdxzjvqo4xpwpjuj4