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.
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Acknowledgements
The author would like to express his deepest thanks to his supervisor, Asuka Takatsu, as well as to Manabu Akaho, Shin-ichi Ohta, Takashi Sakai, Homare Tadano who contributed their support. The author declares that no funds, grants, or other support were received during the preparation of this manuscript.
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Ito, T. Quantitative Estimate of Diameter for Weighted Manifolds Under Integral Curvature Bounds and \(\varepsilon \)-Range. Results Math 77, 221 (2022). https://doi.org/10.1007/s00025-022-01750-0
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DOI: https://doi.org/10.1007/s00025-022-01750-0