This paper is the sequel to our study of heat kernel on Ricci shrinkers [29]. In this paper, we improve many estimates in [29] and extend the recent progress of Bamler [2]. In particular, we drop the compactness and curvature boundedness assumptions and show that the theory of $\mathbb{F}$-convergence holds naturally on any Ricci flows induced by Ricci shrinkers.
Yu Li
,
Bing Wang
. HEAT KERNEL ON RICCI SHRINKERS (II)*[J]. Acta mathematica scientia, Series B, 2024
, 44(5)
: 1639
-1695
.
DOI: 10.1007/s10473-024-0502-7
[1] Andrews B. Gradient and oscillation estimates and their applications in geometric PDE// International Congress of Chinese Mathematicians (ICCM2012), American Mathematical Society, USA, 2012: 3-19
[2] Bamler R H.Entropy and heat kernel bounds on a Ricci flow background. arXiv:2008.07093v3
[3] amler R H. Compactness theory of the space of super Ricci flow. Invent Math, 2023, 233(3): 1121-1277
[4] Bamler R H.Structure theory of non-collapsed limits of Ricci flow. arXiv:2009.03243v2
[5] Bamler R H.On the fundamental group of non-collapsed ancient Ricci flows. arXiv:2110.02254
[6] Bamler R H, Zhang Q S. Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature. Adv Math, 2017, 319: 396-450
[7] Brendle S. Two-point functions and their applications in geometry. Bull Amer Math Soc, 2014, 51(4): 581-596
[8] Cao H D, Chen B L, Zhu X P.Recent developments on Hamilton's Ricci flow// Surveys in Differential Geometry, Vol. 12. Somerville, MA: International Press, 2008: 47-112
[9] Cao H D, Zhou D. On complete gradient shrinking Ricci solitons. J Differ Geom, 2010, 85(2): 175-186
[10] Chan P, Ma Z, Zhang Y.Ancient Ricci flows with asymptotic solitons. arXiv: 2106.06904v1
[11] Cheeger J, Colding T H. On the structure of space with Ricci curvature bounded below I. J Differ Geom, 1997, 46: 406-480
[12] Cheeger J, Naber A. Lower bounds on Ricci curvature and quantitative behavior of singular sets. Invent Math, 2013, 191(2): 321-339
[13] Cheeger J, Naber A. Regularity of Einstein manifolds and the codimension 4 conjecture. Ann Math, 2015, 182(3): 1093-1165
[14] Chen B L. Strong uniqueness of the Ricci flow. J Differ Geom, 2009, 82(2): 363-382
[15] Chen X, Wang B. Space of Ricci flows (I). Comm Pure Appl Math, 2012, 65(10): 1399-1457
[16] Chen X, Wang B. Space of Ricci flows (II)--Part B: Weak compactness of the flows. J Differential Geom, 2020, 116(1): 1-123
[17] Chen X, Wang B. Remarks of weak-compactness along Kähler Ricci flow// Proceedings of the Seventh International Congress of Chinese Mathematicians, ALM 44, 2016: 203-233
[18] Chen C W, Zhang Z. On shrinking gradient Ricci solitons with positive Ricci curvature and small Weyl tensor. Acta Math Sci, 2019, 39B(5): 1235-1239
[19] Chow B, Chu S C, Glickenstein D, et al.The Ricci Flow: Techniques and Applications. Part II. Analytic Aspects. Mathematical Surveys and Monographs, vol 135. Providence RI: American Mathematical Society, 2008
[20] Enders J, Müller R, Topping P. On Type-I singularities in Ricci flow. Comm Anal Geom, 2011, 19(5): 905-922
[21] Friedman A.Partial Differential Equations of Parabolic Type. Mineola, NY: Courier Dover Publications, 2008
[22] Hamilton R S.The formation of singularities in the Ricci flow// Surveys in Differential Geom. Somerville, MA: International Press, 1995: 7-136
[23] Haslhofer R, Müller R. A compactness theorem for complete Ricci shrinkers. Geom Funct Anal, 2011, 21: 1091-1116
[24] Hein H J, Naber A. New logarithmic Sobolev inequalities and an $\epsilon $-regularity theorem for the Ricci flow. Comm Pure Appl Math, 2014, 67(9): 1543-1561
[25] Li H, Li Y, Wang B. On the structure of Ricci shrinkers. J Funct Anal, 2021, 280(9): 108955
[26] Li X, Ni L. Kähler-Ricci shrinkers and ancient solutions with nonnegative orthogonal bisectional curvature. Math Pures Appl, 2020, 138: 28-45
[27] Li X, Ni L, Wang K. Four-dimensional gradient shrinking solitons with positive isotropic curvature. Int Math Res Not, 2018, 2018(3): 949-959
[28] Li Y, Wang B. The rigidity of Ricci shrinkers of dimension four. Trans Amer Math Soc, 2019, 371(10): 6949-6972
[29] Li Y, Wang B. Heat kernel on Ricci shrinkers. Calc Var Partial Differ Equ, 2020, 59: Art 194
[30] Munteanu O, Wang J. Positively curved shrinking Ricci solitons are compact. J Differ Geom, 2017, 106(3): 499-505
[31] Ma Z, Zhang Y. Perelman's entropy on ancient Ricci flows. J Funct Anal2021, 281(9): Art 109195
[32] Naber A. Noncompact shrinking four solitons with nonnegative curvature. J Reine Angew Math, 2010, 645: 125-153
[33] Naff K.Shrinking Ricci solitons with positive isotropic curvature. arXiv: 1905.10305
[34] Ni L, Wallach N. On a classification of gradient shrinking solitons. Math Res Lett, 2008, 15(5): 941-955
[35] Perelman G.The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159
[36] Villani C. Optimal Transport, Old and New. Grundlehren Math Wiss, vol 338. Berlin: Springer, 2008