Abstract
In this article, we consider two different monotone quantities defined for the Ricci flow and show that their asymptotic limits coincide for any ancient solutions. One of the quantities we consider here is Perelman’s reduced volume, while the other is the local quantity discovered by Ecker, Knopf, Ni, and Topping. This establishes a relation between these two monotone quantities.
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References
Carrillo, J., Ni, L.: Sharp logarithmic Sobolev inequalities on gradient solitons and applications. arXiv:0806.2417
Chen B.-L.: Strong uniqueness of the Ricci flow. J. Differ. Geom. 82(2), 363–382 (2009)
Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci Flow. Graduate Studies in Mathematics, vol. 77. American Mathematical Society/Science Press, Providence/New York, (2006)
Chow, B., Chu, S.-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F., Ni, L.: The Ricci Flow: Techniques and Applications. Part I. Geometric Aspects, Mathematical Surveys and Monographs, vol. 135. American Mathematical Society, Providence, RI, (2007)
Ecker, K.: Regularity Theory for Mean Curvature Flow. Progress in Nonlinear Differential Equations and their Applications, vol. 57. Birkhäuser Boston, Inc., Boston, MA (2004)
Ecker K., Knopf D., Ni L., Topping P.: Local monotonicity and mean value formulas for evolving Riemannian manifolds. J. Reine Angew. Math. 616, 89–130 (2008)
Enders, J.: Reduced distance based at singular time in the Ricci flow. arXiv:0711.0558
Feldman M., Ilmanen T., Ni L.: Entropy and reduced distance for Ricci expanders. J. Geom. Anal. 15(1), 49–62 (2005)
Lott J.: Optimal transport and Perelman’s reduced volume. Calc. Var. Partial Differ. Equ. 36(1), 49–84 (2009)
McCann, R., Topping, P.: Ricci flow, entropy and optimal transportation, Amer. J. Math. (to appear)
Müller, R.: Differential Harnack Inequalities and the Ricci Flow. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich (2006)
Naber, A.: Noncompact shrinking 4-solitons with nonnegative curvature. arXiv:0710.5579
Ni, L.: The entropy formula for linear heat equation. J. Geom. Anal. 14(1), 87–100; Addenda: J. Geom. Anal. 14(2), 369–374 (2004)
Ni L.: A matrix Li–Yau–Hamilton estimate for Kähler-Ricci flow. J. Differ. Geom. 75(2), 303–358 (2007)
Ni L.: Mean value theorems on manifolds. Asian J. Math. 11(2), 277–304 (2007)
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159
Topping, P.: \({\mathcal {L}}\) -optimal transportation for Ricci flow, J. Reine Angew. Math. (to appear)
Watson M.-A.: A theory of subtemperatures in several variables. Proc. Lond. Math. Soc. 26(3), 385–417 (1973)
Ye R.: On the l-function and the reduced volume of Perelman I. Trans. Am. Math. Soc. 360(1), 507–531 (2008)
Yokota T.: Perelman’s reduced volume and a gap theorem for the Ricci flow. Commun. Anal. Geom. 17(2), 227–263 (2009)
Yokota, T.: A gap theorem for ancient solutions to the Ricci flow. Proceedings of the 1st MSJ-SI “Probabilistic Approach to Geometry” (to appear)
Zhang Z.-H.: On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc. 137(8), 2755–2759 (2009)
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Yokota, T. On the asymptotic reduced volume of the Ricci flow. Ann Glob Anal Geom 37, 263–274 (2010). https://doi.org/10.1007/s10455-009-9184-6
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DOI: https://doi.org/10.1007/s10455-009-9184-6