Abstract
In this paper, we prove a backward uniqueness theorem for solutions to the inverse mean curvature flow on a closed manifold. As a consequence, the isometry group of a solution cannot expand within the lifetime of the solution.
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References
Daskalopoulos, P., Huisken, G.: Inverse mean curvature evolution of entire graph. Calc. Var. Partial Differ. Equ. 61(2), 53 (2022)
Gerhardt, C.: Flow of nonconvex hypersurfaces into spheres. J. Differ. Geom. 32(1), 299–314 (1990)
Ho, P.T.: Backwards uniqueness of the Yamabe flow. Differ. Geom. Appl. 62, 184–189 (2019)
Huang, H.: Backwards uniqueness of the mean curvature flow. Geom. Dedicata 203, 67–71 (2019)
Huisken, G., Ilmanen, T.: Higher regularity of the inverse mean curvature flow. J. Differ. Geom. 80(3), 433–451 (2008)
Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom. 59(3), 353–437 (2001)
Kotschwar, B.: Backwards uniqueness for the Ricci flow. Int. Math. Res. Not. IMRN 2010(21), 4064–4097 (2010)
Kotschwar, B.: A short proof of backward uniqueness for some geometric evolution equations. Int. J. Math. 27(12), 1650102 (2016)
Lee, M.C., Ma, J.: Uniqueness theorems for non-compact mean curvature flow with possibly unbounded curvatures. Commun. Anal. Geom. 29(6), 1475–1508 (2021)
Urbas, J.: On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures. Math. Z. 205(1), 355–372 (1990)
Zhang, Z.: A note on the backwards uniqueness of the mean curvature flow. Sci. China Math. 62(9), 1793–1798 (2019)
Acknowledgements
The second author was supported by NRF grant funded by MSIT (No. NRF-2020R1A2C1A01005698 and NRF-2021R1A4A1032418).
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Ho, P.T., Pyo, J. Backward uniqueness for the inverse mean curvature flow. Geom Dedicata 217, 41 (2023). https://doi.org/10.1007/s10711-023-00781-3
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DOI: https://doi.org/10.1007/s10711-023-00781-3