Abstract
We construct a sequence of smooth Ricci flows on \(T^2\), with standard uniform C/t curvature decay, and with initial metrics converging to the standard flat unit-area square torus \(g_0\) in the Gromov-Hausdorff sense, with the property that the flows themselves converge not to the static Ricci flow \(g(t)\equiv g_0\), but to the static Ricci flow \(g(t)\equiv 2g_0\) of twice the area.
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References
Bamler, R., Cabezas-Rivas, E., Wilking, B.: The Ricci flow under almost non-negative curvature conditions. Inventiones 217, 95–126 (2019). http://arxiv.org/abs/1707.03002
Deruelle, A.: Private communication (2014)
Deruelle, A., Schulze, F., Simon, M.: On the regularity of Ricci flows coming out of metric spaces. http://arxiv.org/abs/1904.11870
Giesen, G., Topping, P.M.: Existence of Ricci flows of incomplete surfaces. Commun. Partial Differ. Equ. 36, 1860–1880 (2011)
Giesen, G.: Instantaneously complete Ricci flows on surfaces. Ph.D. thesis, University of Warwick (2012)
Hamilton, R.S.: The Ricci flow on surfaces. Mathematics and general relativity (Santa Cruz, CA, : 71 Contemporary Mathematics, 237–262, p. 1988. American Mathematical Society, Providence, RI (1986)
Hamilton, R.S.: The formation of singularities in the Ricci flow. Surveys in differential geometry, Vol. II Cambridge, MA, : 7–136, p. 1995. Press, Cambridge, MA, Internat (1993)
Hochard, R.: Short-time existence of the Ricci flow on complete, non-collapsed \(3\)-manifolds with Ricci curvature bounded from below. http://arxiv.org/abs/1603.08726v1
Hochard, R.: Théorèmes d’existence en temps court du flot de Ricci pour des variétés non-complètes, non-éffondrées, à courbure minorée. Ph.D. thesis, Université de Bordeaux (2019)
Lai, Y.: Ricci flow under local almost non-negative curvature conditions., Adv. Math. 343, 353–392 (2019). https://doi.org/10.1016/j.aim.2018.11.006
McLeod, A.D., Topping: Global regularity of three-dimensional Ricci limit spaces. T.A.M.S. (Series B) (To appear). http://arxiv.org/abs/1803.00414
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. http://arXiv.org/abs/math/0211159v1 (2002)
Richard, T.: Flot de Ricci sans borne supérieure sur la courbure et géométrie de certains espaces métriques. Université de Grenoble, Thesis (2012)
Simon, M.: Ricci flow of non-collapsed three manifolds whose Ricci curvature is bounded from below. Journal für die reine und angewandte Mathematik. 662, 59–94 (2012)
Simon, M., Topping, P.M.: Local control on the geometry in 3D Ricci flow. J. Differ. Geom. (To appear). http://arxiv.org/abs/1611.06137
Simon, M., Topping, P.M.: Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces. Geom. Topol. (To appear). http://arxiv.org/abs/1706.09490
Topping, P.M. : Lectures on the Ricci flow. L.M.S. Lect. Notes Ser. 325 C.U.P. (2006). http://www.warwick.ac.uk/~maseq/RFnotes.html
Topping, P.M.: Uniqueness of instantaneously complete Ricci flows. Geom. Topol. 19, 1477–1492 (2015)
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Topping, P.M. (2021). Loss of Initial Data Under Limits of Ricci Flows. In: Hoffmann, T., Kilian, M., Leschke, K., Martin, F. (eds) Minimal Surfaces: Integrable Systems and Visualisation. m:iv m:iv m:iv m:iv 2017 2018 2018 2019. Springer Proceedings in Mathematics & Statistics, vol 349. Springer, Cham. https://doi.org/10.1007/978-3-030-68541-6_15
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