Abstract
By using the nice behavior of the Hawking mass of the slices of a weak solution of inverse mean curvature flow in three-dimensional asymptotically hyperbolic manifolds, we are able to show that each slice of the flow is star-shaped after a long time, and then we get the regularity of the weak solution of inverse mean curvature flow in asymptotically hyperbolic manifolds. As an application, we prove that the limit of the Hawking mass of the slices of a weak solution of inverse mean curvature flow with any connected C2-smooth surface as initial data in asymptotically anti-de Sitter-Schwarzschild manifolds with positive mass is greater than or equal to the total mass, which is completely different from the situation in the asymptotically flat case.
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 11671015 and 11731001). The authors thank the referees for their careful reading and helpful suggestions on the statements of this paper.
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Shi, Y., Zhu, J. Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds with dimension 3. Sci. China Math. 64, 1109–1126 (2021). https://doi.org/10.1007/s11425-020-1860-6
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DOI: https://doi.org/10.1007/s11425-020-1860-6