A note on convergence of noncompact nonsingular solutions of the Ricci flow

Zhang, Qi

doi:10.1090/proc/15813

A note on convergence of noncompact nonsingular solutions of the Ricci flow
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Proc. Amer. Math. Soc. 150 (2022), 3627-3634 Request permission

Abstract:

We extend some convergence results on nonsingular compact Ricci flows in the papers by Hamilton [Comm. Anal. Geom. 7 (1999), pp. 695–729], Sesum [Math. Res. Lett. 12 (2005), pp. 623–632] and Fang, Zhang, and Zhang [J. Geom. Anal. 20 (2010), pp. 592–608] to certain infinite volume noncompact cases which are “partially” nonsingular. As an application, for a finite time singularity which is partially type I, it is shown that a blow up limit is a gradient shrinking soliton.

References

Richard H. Bamler, Structure theory of non-collapsed limits of Ricci flows, arXiv:2009.03243, 2020.
Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci flow: techniques and applications. Part III. Geometric-analytic aspects, Mathematical Surveys and Monographs, vol. 163, American Mathematical Society, Providence, RI, 2010. MR 2604955, DOI 10.1090/surv/163
Jeff Cheeger, Mikhail Gromov, and Michael Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geometry 17 (1982), no. 1, 15–53. MR 658471
Siu Yuen Cheng, Peter Li, and Shing Tung Yau, On the upper estimate of the heat kernel of a complete Riemannian manifold, Amer. J. Math. 103 (1981), no. 5, 1021–1063. MR 630777, DOI 10.2307/2374257
Xiaodong Cao and Qi S. Zhang, The conjugate heat equation and ancient solutions of the Ricci flow, Adv. Math. 228 (2011), no. 5, 2891–2919. MR 2838064, DOI 10.1016/j.aim.2011.07.022
Joerg Enders, Reto Müller, and Peter M. Topping, On type-I singularities in Ricci flow, Comm. Anal. Geom. 19 (2011), no. 5, 905–922. MR 2886712, DOI 10.4310/CAG.2011.v19.n5.a4
Fuquan Fang, Yuguang Zhang, and Zhenlei Zhang, Non-singular solutions to the normalized Ricci flow equation, Math. Ann. 340 (2008), no. 3, 647–674. MR 2357999, DOI 10.1007/s00208-007-0164-5
Fuquan Fang, Zhenlei Zhang, and Yuguang Zhang, Non-singular solutions of normalized Ricci flow on noncompact manifolds of finite volume, J. Geom. Anal. 20 (2010), no. 3, 592–608. MR 2610891, DOI 10.1007/s12220-010-9120-9
Richard S. Hamilton, Non-singular solutions of the Ricci flow on three-manifolds, Comm. Anal. Geom. 7 (1999), no. 4, 695–729. MR 1714939, DOI 10.4310/CAG.1999.v7.n4.a2
Richard S. Hamilton, A compactness property for solutions of the Ricci flow, Amer. J. Math. 117 (1995), no. 3, 545–572. MR 1333936, DOI 10.2307/2375080
Aaron Naber, Noncompact shrinking four solitons with nonnegative curvature, J. Reine Angew. Math. 645 (2010), 125–153. MR 2673425, DOI 10.1515/CRELLE.2010.062
Perelman, Grisha, The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159, 2002.
Grisha Perelman, Ricci flow with surgery on three-manifolds, arXiv:math/0303109, 2003.
O. S. Rothaus, Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators, J. Functional Analysis 42 (1981), no. 1, 110–120. MR 620582, DOI 10.1016/0022-1236(81)90050-1
Sesum, N. Convergence of a Ricci flow towards a Ricci soliton; Comm. Anal. Geom. 14 (2006), no.2, 283-343;
Luen-Fai Tam, Exhaustion functions on complete manifolds, Recent advances in geometric analysis, Adv. Lect. Math. (ALM), vol. 11, Int. Press, Somerville, MA, 2010, pp. 211–215. MR 2648946
Qi S. Zhang, Extremal of log Sobolev inequality and $W$ entropy on noncompact manifolds, J. Funct. Anal. 263 (2012), no. 7, 2051–2101. MR 2956934, DOI 10.1016/j.jfa.2012.07.005