Abstract
In this paper, we study the classification of \(\kappa \)-noncollapsed ancient solutions to three-dimensional Ricci flow on \(S^3\). We prove that such a solution is either isometric to a family of shrinking round spheres, or the Type II ancient solution constructed by Perelman.
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Angenent, S., Brendle, S., Daskalopoulos, P., Šešum, N.: Unique asymptotics of compact ancient solutions to three-dimensional Ricci flow, to appear in Comm. Pure Appl. Math
Angenent, S., Daskalopoulos, P., Šešum, N.: Unique asymptotics of ancient convex mean curvature flow solutions. J. Diff. Geom. 111, 381–455 (2019)
Angenent, S., Daskalopoulos, P., Šešum, N.: Uniqueness of two-convex closed ancient solutions to the mean curvature flow. Ann. Math. 192, 353–436 (2020)
Bamler, R., Kleiner, B.: On the rotational symmetry of \(3\)-dimensional \(\kappa \)-solutions, arxiv:1904.05388
Bourni, T., Langford, M., Tinaglia, G.: Collapsing ancient solutions of mean curvature flow, arxiv:1705.06981
Brendle, S.: Rotational symmetry of self-similar solutions to the Ricci flow. Invent. Math. 194, 731–764 (2013)
Brendle, S.: Ancient solutions to the Ricci flow in dimension \(3\). Acta Math. 225, 1–102 (2020)
Brendle, S., Choi, K.: Uniqueness of convex ancient solutions to mean curvature flow in \(\mathbb{R}^3\). Invent. Math. 217, 35–76 (2019)
Brendle, S., Choi, K.: Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions, to appear in Geom. Topol
Brendle, S., Huisken, G., Sinestrari, C.: Ancient solutions to the Ricci flow with pinched curvature. Duke Math. J. 158, 537–551 (2011)
Bryant, R.L.: Ricci flow solitons in dimension three with \(SO(3)\)-symmetries, available at www.math.duke.edu/~bryant/3DRotSymRicciSolitons.pdf
Daskalopoulos, P., Hamilton, R., Šešum, N.: Classification of compact ancient solutions to the curve shortening flow. J. Diff. Geom. 84, 455–464 (2010)
Daskalopoulos, P., Hamilton, R., Šešum, N.: Classification of ancient compact solutions to the Ricci flow on surfaces. J. Diff. Geom. 91, 171–214 (2012)
Fateev, V.A., Onofri, E., Zamolodchikov, A.B.: Integrable deformations of the O(3) sigma model. The sausage model. Nuclear Phys. B406, 521–565 (1993)
Hamilton, R.: The Harnack estimate for the Ricci flow. J. Diff. Geom. 37, 225–243 (1993)
Haslhofer, R., Hershkovits, O.: Ancient solutions of the mean curvature flow. Commun. Anal. Geom. 24, 593–604 (2016)
King, J.R.: Exact polynomial solutions to some nonlinear diffusion equations. Phys. D 64, 39–65 (1993)
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159
Perelman, G.: Ricci flow with surgery on three-manifolds, arXiv:math/0303109
Rosenau, P.: Fast and super fast diffusion processes. Phys. Rev. Lett. 74, 1056–1059 (1995)
White, B.: The nature of singularities in mean curvature flow of mean convex sets. J. Am. Math. Soc. 16, 123–138 (2003)
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The first author was supported by the National Science Foundation under grant DMS-1806190 and by the Simons Foundation. The second author was supported by the National Science Foundation under grant DMS-1266172. The third author was supported by the National Science Foundation under grants DMS-1056387 and DMS-1811833.
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Brendle, S., Daskalopoulos, P. & Sesum, N. Uniqueness of compact ancient solutions to three-dimensional Ricci flow. Invent. math. 226, 579–651 (2021). https://doi.org/10.1007/s00222-021-01054-0
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DOI: https://doi.org/10.1007/s00222-021-01054-0