The Kähler–Ricci flow around complete bounded curvature Kähler metrics

Chau, Albert; Lee, Man-Chun

doi:10.1090/tran/8015

The Kähler–Ricci flow around complete bounded curvature Kähler metrics
HTML articles powered by AMS MathViewer

by Albert Chau and Man-Chun Lee PDF

Trans. Amer. Math. Soc. 373 (2020), 3627-3647 Request permission

Abstract:

We produce complete bounded curvature solutions to the Kähler–Ricci flow with existence time estimates, assuming only that the initial data is a smooth Kähler metric uniformly equivalent to another complete bounded curvature Kähler metric. We obtain related flow results for nonsmooth as well as degenerate initial conditions. We also obtain a stability result for complex space forms under the flow.

References

Huai-Dong Cao, Bing-Long Chen, and Xi-Ping Zhu, Recent developments on Hamilton’s Ricci flow, Surveys in differential geometry. Vol. XII. Geometric flows, Surv. Differ. Geom., vol. 12, Int. Press, Somerville, MA, 2008, pp. 47–112. MR 2488948, DOI 10.4310/SDG.2007.v12.n1.a3
Esther Cabezas-Rivas and Burkhard Wilking, How to produce a Ricci flow via Cheeger-Gromoll exhaustion, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 12, 3153–3194. MR 3429162, DOI 10.4171/JEMS/582
Richard H. Bamler, Esther Cabezas-Rivas, and Burkhard Wilking, The Ricci flow under almost non-negative curvature conditions, Invent. Math. 217 (2019), no. 1, 95–126. MR 3958792, DOI 10.1007/s00222-019-00864-7
Albert Chau, Ka-Fai Li, and Luen-Fai Tam, Deforming complete Hermitian metrics with unbounded curvature, Asian J. Math. 20 (2016), no. 2, 267–292. MR 3480020, DOI 10.4310/AJM.2016.v20.n2.a3
Albert Chau, Ka-Fai Li, and Luen-Fai Tam, An existence time estimate for Kähler-Ricci flow, Bull. Lond. Math. Soc. 48 (2016), no. 4, 699–707. MR 3532144, DOI 10.1112/blms/bdw035
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 II, Mathematical Surveys and Monographs, vol. 144, American Mathematical Society, Providence, RI, 2008. Analytic aspects. MR 2365237, DOI 10.1090/surv/144
Gregor Giesen and Peter M. Topping, Ricci flow of negatively curved incomplete surfaces, Calc. Var. Partial Differential Equations 38 (2010), no. 3-4, 357–367. MR 2647124, DOI 10.1007/s00526-009-0290-x
Gregor Giesen and Peter M. Topping, Existence of Ricci flows of incomplete surfaces, Comm. Partial Differential Equations 36 (2011), no. 10, 1860–1880. MR 2832165, DOI 10.1080/03605302.2011.558555
Gregor Giesen and Peter M. Topping, Ricci flows with unbounded curvature, Math. Z. 273 (2013), no. 1-2, 449–460. MR 3010170, DOI 10.1007/s00209-012-1014-z
H. Ge, A. Lin, and L.-M. Shen, The Kähler-Ricci flow on pseudoconvex domain, arXiv: 1803.07761.
S.-C. Huang, M.-C. Lee, L.-F. Tam, and F. Tong, Longtime existence of Kähler–Ricci flow and holomorphic sectional curvature, arXiv:1805.12328.
Shaochuang Huang, Man-Chun Lee, and Luen-Fai Tam, Instantaneously complete Chern-Ricci flow and Kähler-Einstein metrics, Calc. Var. Partial Differential Equations 58 (2019), no. 5, Paper No. 161, 34. MR 4010637, DOI 10.1007/s00526-019-1612-2
F. He, Existence and applications of Ricci flows via pseudolocality, arXiv:1610.01735 (2016).
F. He and M.-C. Lee, Weakly PIC1 manifolds with maximal volume growth, arXiv:1811.03318.
R. Hochard, Short-time existence of the Ricci flow on complete, non-collapsed 3-manifolds with Ricci curvature bounded from below, arXiv:1603.08726 (2016).
Lawrence C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math. 35 (1982), no. 3, 333–363. MR 649348, DOI 10.1002/cpa.3160350303
Yi Lai, Ricci flow under local almost non-negative curvature conditions, Adv. Math. 343 (2019), 353–392. MR 3881661, DOI 10.1016/j.aim.2018.11.006
M.-C. Lee and L.-F. Tam, Chern-Ricci flows on noncompact manifolds., arXiv:1708.00141. To appear in J. Differential Geom.
M.-C. Lee, Noncompact Hermitian flow and applications, arXiv:1812.04610.
John Lott and Zhou Zhang, Ricci flow on quasiprojective manifolds II, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 8, 1813–1854. MR 3519542, DOI 10.4171/JEMS/630
N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 3, 487–523, 670 (Russian). MR 661144
G. Perelman, The entropy formula for the Ricci flow and its geometric applications, http://arXiv.org/abs/math/0211159v1 (2002).
Wan-Xiong Shi, Ricci deformation of the metric on complete noncompact Riemannian manifolds, J. Differential Geom. 30 (1989), no. 2, 303–394. MR 1010165
Wan-Xiong Shi, Ricci flow and the uniformization on complete noncompact Kähler manifolds, J. Differential Geom. 45 (1997), no. 1, 94–220. MR 1443333
Miles Simon, Deformation of $C^0$ Riemannian metrics in the direction of their Ricci curvature, Comm. Anal. Geom. 10 (2002), no. 5, 1033–1074. MR 1957662, DOI 10.4310/CAG.2002.v10.n5.a7
Oliver C. Schnürer, Felix Schulze, and Miles Simon, Stability of Euclidean space under Ricci flow, Comm. Anal. Geom. 16 (2008), no. 1, 127–158. MR 2411470
Oliver C. Schnürer, Felix Schulze, and Miles Simon, Stability of hyperbolic space under Ricci flow, Comm. Anal. Geom. 19 (2011), no. 5, 1023–1047. MR 2886716, DOI 10.4310/CAG.2011.v19.n5.a8
M. Simon and P.-M. Topping, Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces, arXiv:1706.09490 (2017).
M. Simon and P.-M. Topping, Local control on the geometry in 3D Ricci flow, arXiv:1611.06137 (2016).
Morgan Sherman and Ben Weinkove, Interior derivative estimates for the Kähler-Ricci flow, Pacific J. Math. 257 (2012), no. 2, 491–501. MR 2972475, DOI 10.2140/pjm.2012.257.491
Jian Song and Ben Weinkove, An introduction to the Kähler-Ricci flow, An introduction to the Kähler-Ricci flow, Lecture Notes in Math., vol. 2086, Springer, Cham, 2013, pp. 89–188. MR 3185333, DOI 10.1007/978-3-319-00819-6_{3}
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
Guoyi Xu, Short-time existence of the Ricci flow on noncompact Riemannian manifolds, Trans. Amer. Math. Soc. 365 (2013), no. 11, 5605–5654. MR 3091259, DOI 10.1090/S0002-9947-2013-05998-3

Additional Information

Albert Chau
Affiliation: Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, B.C., Canada V6T 1Z2
MR Author ID: 749289
Email: chau@math.ubc.ca
Man-Chun Lee
Affiliation: Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, B.C., Canada V6T 1Z2
Address at time of publication: Department of Mathematics, Room 225, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208
MR Author ID: 1322380
Email: mclee@math.northwestern.edu
Received by editor(s): May 30, 2019
Received by editor(s) in revised form: September 17, 2019
Published electronically: February 21, 2020
Additional Notes: The first author’s research was partially supported by NSERC grant no. #327637-06.
Journal: Trans. Amer. Math. Soc. 373 (2020), 3627-3647
DOI: https://doi.org/10.1090/tran/8015
MathSciNet review: 4082250