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Type II singularities on complete non-compact Yamabe flow

Beomjun Choi, Panagiota Daskalopoulos and John King

Abstract

This work concerns with the existence and detailed asymptotic analysis of type II singularities for solutions to complete non-compact conformally flat Yamabe flow with cylindrical behavior at infinity. We provide the specific blow-up rate of the maximum curvature and show that the solution converges, after blowing-up around the curvature maximum points, to a rotationally symmetric steady soliton. It is the first time that the steady soliton is shown to be a finite time singularity model of the Yamabe flow.

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1600658

Award Identifier / Grant number: DMS-1600658

Funding statement: Beomjun Choi has been partially supported by NSF grant DMS-1600658. Panagiota Daskalopoulos has been partially supported by NSF grant DMS-1600658 and the Simons Foundation.

Acknowledgements

The authors are indebted to Jim Isenberg and Mariel Sáez for useful discussions about this work.

References

[1] S. B. Angenent, J. Isenberg and D. Knopf, Formal matched asymptotics for degenerate Ricci flow neckpinches, Nonlinearity 24 (2011), no. 8, 2265–2280. 10.1088/0951-7715/24/8/007Search in Google Scholar

[2] S. B. Angenent, J. Isenberg and D. Knopf, Degenerate neckpinches in Ricci flow, J. reine angew. Math. 709 (2015), 81–117. 10.1515/crelle-2013-0105Search in Google Scholar

[3] S. B. Angenent and J. J. L. Velázquez, Degenerate neckpinches in mean curvature flow, J. reine angew. Math. 482 (1997), 15–66. Search in Google Scholar

[4] S. Brendle, Convergence of the Yamabe flow for arbitrary initial energy, J. Differential Geom. 69 (2005), no. 2, 217–278. 10.4310/jdg/1121449107Search in Google Scholar

[5] S. Brendle, Convergence of the Yamabe flow in dimension 6 and higher, Invent. Math. 170 (2007), no. 3, 541–576. 10.1007/s00222-007-0074-xSearch in Google Scholar

[6] H.-D. Cao, X. Sun and Y. Zhang, On the structure of gradient Yamabe solitons, Math. Res. Lett. 19 (2012), no. 4, 767–774. 10.4310/MRL.2012.v19.n4.a3Search in Google Scholar

[7] G. Catino, C. Mantegazza and L. Mazzieri, On the global structure of conformal gradient solitons with nonnegative Ricci tensor, Commun. Contemp. Math. 14 (2012), no. 6, Article ID 1250045. 10.1142/S0219199712500459Search in Google Scholar

[8] B. Choi and P. Daskalopoulos, Yamabe flow: Steady solitons and type II singularities, Nonlinear Anal. 173 (2018), 1–18. 10.1016/j.na.2018.03.008Search in Google Scholar

[9] B. Chow, The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature, Comm. Pure Appl. Math. 45 (1992), no. 8, 1003–1014. 10.1002/cpa.3160450805Search in Google Scholar

[10] P. Daskalopoulos, J. King and N. Sesum, Extinction profile of complete non-compact solutions to the Yamabe flow, Comm. Anal. Geom. 27 (2019), no. 8, 1757–1798. 10.4310/CAG.2019.v27.n8.a4Search in Google Scholar

[11] P. Daskalopoulos and N. Sesum, On the extinction profile of solutions to fast diffusion, J. reine angew. Math. 622 (2008), 95–119. 10.1515/CRELLE.2008.066Search in Google Scholar

[12] P. Daskalopoulos and N. Sesum, The classification of locally conformally flat Yamabe solitons, Adv. Math. 240 (2013), 346–369. 10.1016/j.aim.2013.03.011Search in Google Scholar

[13] R. S. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986), no. 2, 153–179. 10.4310/jdg/1214440433Search in Google Scholar

[14] R. S. Hamilton, Lectures on geometric flows, unpublished, 1989. Search in Google Scholar

[15] S.-Y. Hsu, Singular limit and exact decay rate of a nonlinear elliptic equation, Nonlinear Anal. 75 (2012), no. 7, 3443–3455. 10.1016/j.na.2012.01.009Search in Google Scholar

[16] J. Isenberg and H. Wu, Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up, J. reine angew. Math. 754 (2019), 225–251. 10.1515/crelle-2017-0019Search in Google Scholar

[17] H. Schwetlick and M. Struwe, Convergence of the Yamabe flow for “large” energies, J. reine angew. Math. 562 (2003), 59–100. 10.1515/crll.2003.078Search in Google Scholar

[18] H. Wu, On Type-II singularities in Ricci flow on N , Comm. Partial Differential Equations 39 (2014), no. 11, 2064–2090. 10.1080/03605302.2014.931097Search in Google Scholar

[19] R. Ye, Global existence and convergence of Yamabe flow, J. Differential Geom. 39 (1994), no. 1, 35–50. 10.4310/jdg/1214454674Search in Google Scholar

Received: 2018-09-28
Published Online: 2020-10-08
Published in Print: 2021-03-01

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