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Global viscosity solutions of generalized Kähler-Ricci flow


Author: Jeffrey Streets
Journal: Proc. Amer. Math. Soc. 146 (2018), 747-757
MSC (2010): Primary 53C44, 35D40
DOI: https://doi.org/10.1090/proc/13772
Published electronically: August 30, 2017
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Abstract: We apply ideas from viscosity theory to establish the existence of a unique global weak solution to the generalized Kähler-Ricci flow in the setting of commuting complex structures. Our results are restricted to the case of a smooth manifold with smooth background data. We discuss the possibility of extending these results to more singular settings, pointing out a key error in the existing literature on viscosity solutions to complex Monge-Ampère equations/Kähler-Ricci flow.


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  • [1] Vestislav Apostolov and Marco Gualtieri, Generalized Kähler manifolds, commuting complex structures, and split tangent bundles, Comm. Math. Phys. 271 (2007), no. 2, 561-575. MR 2287917, https://doi.org/10.1007/s00220-007-0196-4
  • [2] Daniel Azagra, Juan Ferrera, and Beatriz Sanz, Viscosity solutions to second order partial differential equations on Riemannian manifolds, J. Differential Equations 245 (2008), no. 2, 307-336. MR 2428000, https://doi.org/10.1016/j.jde.2008.03.030
  • [3] Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1-67. MR 1118699, https://doi.org/10.1090/S0273-0979-1992-00266-5
  • [4] Philippe Eyssidieux, Vincent Guedj, and Ahmed Zeriahi, Continuous approximation of quasiplurisubharmonic functions, Analysis, complex geometry, and mathematical physics: in honor of Duong H. Phong, Contemp. Math., vol. 644, Amer. Math. Soc., Providence, RI, 2015, pp. 67-78. MR 3372461, https://doi.org/10.1090/conm/644/12787
  • [5] P. Eyssidieux, V. Guedj, and A. Zeriahi, Convergence of weak Kähler-Ricci flows on minimal models of positive Kodaira dimension, arXiv:1604.07001.
  • [6] P. Eyssidieux, V. Guedj, and A. Zeriahi, Erratum to Viscosity solutions to complex Monge-Ampère equations, arXiv:1610.03087.
  • [7] Philippe Eyssidieux, Vincent Guedj, and Ahmed Zeriahi, Viscosity solutions to degenerate complex Monge-Ampère equations, Comm. Pure Appl. Math. 64 (2011), no. 8, 1059-1094. MR 2839271, https://doi.org/10.1002/cpa.20364
  • [8] Philippe Eyssidieux, Vincent Guedj, and Ahmed Zeriahi, Weak solutions to degenerate complex Monge-Ampère flows II, Adv. Math. 293 (2016), 37-80. MR 3474319, https://doi.org/10.1016/j.aim.2016.02.010
  • [9] Hitoshi Ishii, On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs, Comm. Pure Appl. Math. 42 (1989), no. 1, 15-45. MR 973743, https://doi.org/10.1002/cpa.3160420103
  • [10] H. Ishii and P.-L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations 83 (1990), no. 1, 26-78. MR 1031377, https://doi.org/10.1016/0022-0396(90)90068-Z
  • [11] S. J. Gates Jr., C. M. Hull, and M. Roček, Twisted multiplets and new supersymmetric nonlinear $ \sigma$-models, Nuclear Phys. B 248 (1984), no. 1, 157-186. MR 776369, https://doi.org/10.1016/0550-3213(84)90592-3
  • [12] Marco Gualtieri, Generalized complex geometry, Ann. of Math. (2) 174 (2011), no. 1, 75-123. MR 2811595, https://doi.org/10.4007/annals.2011.174.1.3
  • [13] Nigel Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math. 54 (2003), no. 3, 281-308. MR 2013140, https://doi.org/10.1093/qjmath/54.3.281
  • [14] Lars Hörmander, Notions of convexity, Progress in Mathematics, vol. 127, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1301332
  • [15] C. Lu and V. Nguyen, Degenerate complex Hessian equations on compact Kähler manifolds, arXiv:1402.5147.
  • [16] J. Streets, Pluriclosed flow on generalized Kähler manifolds with split tangent bundle, arXiv:1405.0727, to appear Crelles Journal
  • [17] Jeffrey Streets, Pluriclosed flow, Born-Infeld geometry, and rigidity results for generalized Kähler manifolds, Comm. Partial Differential Equations 41 (2016), no. 2, 318-374. MR 3462132, https://doi.org/10.1080/03605302.2015.1116560
  • [18] Jeffrey Streets and Gang Tian, Generalized Kähler geometry and the pluriclosed flow, Nuclear Phys. B 858 (2012), no. 2, 366-376. MR 2881439, https://doi.org/10.1016/j.nuclphysb.2012.01.008
  • [19] Jeffrey Streets and Micah Warren, Evans-Krylov estimates for a nonconvex Monge-Ampère equation, Math. Ann. 365 (2016), no. 1-2, 805-834. MR 3498927, https://doi.org/10.1007/s00208-015-1293-x
  • [20] Gang Tian and Zhou Zhang, On the Kähler-Ricci flow on projective manifolds of general type, Chinese Ann. Math. Ser. B 27 (2006), no. 2, 179-192. MR 2243679, https://doi.org/10.1007/s11401-005-0533-x
  • [21] Shing Tung Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339-411. MR 480350, https://doi.org/10.1002/cpa.3160310304

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Additional Information

Jeffrey Streets
Affiliation: Department of Mathematics, Rowland Hall, University of California, Irvine, Irvine, California 92617
Email: jstreets@uci.edu

DOI: https://doi.org/10.1090/proc/13772
Received by editor(s): October 6, 2016
Received by editor(s) in revised form: March 22, 2017
Published electronically: August 30, 2017
Additional Notes: The author gratefully acknowledges support from the NSF via DMS-1454854, and from an Alfred P. Sloan Fellowship.
Communicated by: Lei Ni
Article copyright: © Copyright 2017 American Mathematical Society

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