Abstract
Asymptotically flat manifolds are being studied intensively for the positive mass theorem and the Riemannian Penrose inequality. We obtain an asymptotic relative volume comparison estimate. From the volume comparison, we show the rigidity of total mass with respect to some integral norm of Ricci curvature. Also we show that the integral norm of Ricci curvature has a positive lower bound for an asymptotically flat manifold whose boundary is a compact minimal hypersurface.
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28 August 2019
Dear readers, the author found some minor corrections in his online published article.
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Acknowledgements
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2019R1F1A1042490).
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The original version of this article was revised: Dear readers, The author found some minor corrections in his online published article. In Theorem 4, Proposition 1 and Proposition 2, vol.() should be changed to vol(). In Proposition 1, vol.(B(t+D)) and vol.(B(tD)) should be changed to vol(B(t+D)) and vol(B(tD)), respectively. The original article has been corrected.
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Paeng, SH. Relative volume comparison for asymptotically flat manifolds and rigidity of total mass. Ann Glob Anal Geom 56, 567–580 (2019). https://doi.org/10.1007/s10455-019-09679-4
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DOI: https://doi.org/10.1007/s10455-019-09679-4