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Non-compact \(\text {RCD}(0,N)\) Spaces with Linear Volume Growth

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Abstract

Since non-compact \(\text {RCD}(0, N)\) spaces have at least linear volume growth, we study non-compact \(\text {RCD}(0,N)\) spaces with linear volume growth in this paper. One of the main results is that the diameter of level sets of a Busemann function grows at most linearly on a non-compact \(\text {RCD}(0,N)\) space satisfying the linear volume growth condition. Another main result in this paper is a rigidity theorem at the non-compact end for a \(\text {RCD}(0,N)\) space with strongly minimal volume growth. These results generalize some theorems on non-compact manifolds with non-negative Ricci curvature to non-smooth settings.

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Acknowledgements

The author would like to thank Prof. X.P. Zhu, B.L. Chen, and H.C. Zhang for their encouragements and helpful discussions. The author is grateful to the anonymous referees for careful reading and giving many valuable suggestions.

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  1. Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, People’s Republic of China

    Xian-Tao Huang

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Huang, XT. Non-compact \(\text {RCD}(0,N)\) Spaces with Linear Volume Growth. J Geom Anal 28, 1005–1051 (2018). https://doi.org/10.1007/s12220-017-9852-x

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