Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 1 (2014), 215-226.

The $J$-flow on Kähler surfaces: a boundary case

Hao Fang, Mijia Lai, Jian Song, and Ben Weinkove

Full-text: Open access

Abstract

We study the J-flow on Kähler surfaces when the Kähler class lies on the boundary of the open cone for which global smooth convergence holds and satisfies a nonnegativity condition. We obtain a C0 estimate and show that the J-flow converges smoothly to a singular Kähler metric away from a finite number of curves of negative self-intersection on the surface. We discuss an application to the Mabuchi energy functional on Kähler surfaces with ample canonical bundle.

Article information

Source
Anal. PDE, Volume 7, Number 1 (2014), 215-226.

Dates
Received: 9 January 2013
Accepted: 22 August 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731470

Digital Object Identifier
doi:10.2140/apde.2014.7.215

Mathematical Reviews number (MathSciNet)
MR3219504

Zentralblatt MATH identifier
1294.53060

Subjects
Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]

Keywords
Kähler $J$-flow complex Monge–Ampère

Citation

Fang, Hao; Lai, Mijia; Song, Jian; Weinkove, Ben. The $J$-flow on Kähler surfaces: a boundary case. Anal. PDE 7 (2014), no. 1, 215--226. doi:10.2140/apde.2014.7.215. https://projecteuclid.org/euclid.apde/1513731470


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References

  • T. Aubin, “Équations du type Monge–Ampère sur les variétés Kählériennes compactes”, Bull. Sci. Math. $(2)$ 102:1 (1978), 63–95.
  • S. Boucksom, P. Eyssidieux, V. Guedj, and A. Zeriahi, “Monge–Ampère equations in big cohomology classes”, Acta Math. 205:2 (2010), 199–262.
  • N. Buchdahl, “On compact Kähler surfaces”, Ann. Inst. Fourier $($Grenoble$)$ 49:1 (1999), 287–302. http://msp.org/idx/mr/2000f:32029MR 2000f:32029
  • E. Calabi, “Extremal Kähler metrics”, pp. 259–290 in Seminar on Differential Geometry, edited by S.-T. Yau, Ann. of Math. Stud. 102, Princeton University Press, 1982.
  • X. Chen, “On the lower bound of the Mabuchi energy and its application”, Int. Math. Res. Not. 2000:12 (2000), 607–623.
  • X. Chen, “A new parabolic flow in Kähler manifolds”, Comm. Anal. Geom. 12:4 (2004), 837–852.
  • J.-P. Demailly, T. Peternell, and M. Schneider, “Compact complex manifolds with numerically effective tangent bundles”, J. Algebraic Geom. 3:2 (1994), 295–345.
  • S. K. Donaldson, “Moment maps and diffeomorphisms”, Asian J. Math. 3:1 (1999), 1–15.
  • S. K. Donaldson, “Scalar curvature and stability of toric varieties”, J. Differential Geom. 62:2 (2002), 289–349.
  • L. C. Evans, “Classical solutions of fully nonlinear, convex, second-order elliptic equations”, Comm. Pure Appl. Math. 35:3 (1982), 333–363.
  • P. Eyssidieux, V. Guedj, and A. Zeriahi, “Singular Kähler–Einstein metrics”, J. Amer. Math. Soc. 22:3 (2009), 607–639.
  • P. Eyssidieux, V. Guedj, and A. Zeriahi, “Viscosity solutions to degenerate complex Monge–Ampère equations”, Comm. Pure Appl. Math. 64:8 (2011), 1059–1094.
  • H. Fang and M. Lai, “Convergence of general inverse $\sigma_k$-flow on Kähler manifolds with Calabi ansatz”, preprint, 2012.
  • H. Fang and M. Lai, “On the geometric flows solving Kählerian inverse $\sigma\sb k$ equations”, Pacific J. Math. 258:2 (2012), 291–304.
  • H. Fang, M. Lai, and X. Ma, “On a class of fully nonlinear flows in Kähler geometry”, J. Reine Angew. Math. 653 (2011), 189–220.
  • V. Guedj and A. Zeriahi, “Intrinsic capacities on compact Kähler manifolds”, J. Geom. Anal. 15:4 (2005), 607–639.
  • S. Kołodziej, “The complex Monge–Ampère equation”, Acta Math. 180:1 (1998), 69–117.
  • S. Kołodziej, “The complex Monge–Ampère equation and pluripotential theory”, Mem. Amer. Math. Soc. 178:840 (2005).
  • N. V. Krylov, “\cyr Ogranichenno neodnorodnye e1llipticheskie i parabolicheskie uravneniya”, Izv. Akad. Nauk SSSR Ser. Mat. 46:3 (1982), 487–523. Translated as “Boundedly nonhomogeneous elliptic and parabolic equations” in Math. USSR Izv. 20:3 (1983), 459–492.
  • A. Lamari, “Le cône Kählérien d'une surface”, J. Math. Pures Appl. $(9)$ 78:3 (1999), 249–263.
  • R. Lazarsfeld, Positivity in algebraic geometry, I: Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete $(3)$ 48, Springer, Berlin, 2004.
  • C. LeBrun and S. R. Simanca, “Extremal Kähler metrics and complex deformation theory”, Geom. Funct. Anal. 4:3 (1994), 298–336.
  • T. Mabuchi, “$K$-energy maps integrating Futaki invariants”, Tohoku Math. J. $(2)$ 38:4 (1986), 575–593.
  • D. H. Phong, J. Song, J. Sturm, and B. Weinkove, “The Moser–Trudinger inequality on Kähler–Einstein manifolds”, Amer. J. Math. 130:4 (2008), 1067–1085.
  • J. Ross, “Unstable products of smooth curves”, Invent. Math. 165:1 (2006), 153–162.
  • J. Song and B. Weinkove, “On the convergence and singularities of the $J$-flow with applications to the Mabuchi energy”, Comm. Pure Appl. Math. 61:2 (2008), 210–229.
  • G. Tian, “Kähler–Einstein metrics with positive scalar curvature”, Invent. Math. 130:1 (1997), 1–37.
  • G. Tian, Canonical metrics in Kähler geometry, Birkhäuser, Basel, 2000.
  • G. Tian and X. Zhu, “A nonlinear inequality of Moser–Trudinger type”, Calc. Var. Partial Differential Equations 10:4 (2000), 349–354.
  • H. Tsuji, “Existence and degeneration of Kähler–Einstein metrics on minimal algebraic varieties of general type”, Math. Ann. 281:1 (1988), 123–133.
  • B. Weinkove, “Convergence of the $J$-flow on Kähler surfaces”, Comm. Anal. Geom. 12:4 (2004), 949–965.
  • B. Weinkove, “On the $J$-flow in higher dimensions and the lower boundedness of the Mabuchi energy”, J. Differential Geom. 73:2 (2006), 351–358.
  • S.-T. Yau, “On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation, I”, Comm. Pure Appl. Math. 31:3 (1978), 339–411.
  • S.-T. Yau, “Open problems in geometry”, pp. 1–28 in Differential geometry, 1: Partial differential equations on manifolds (Los Angeles, 1990), edited by R. E. Greene and S.-T. Yau, Proc. Sympos. Pure Math. 54, American Mathematical Society, Providence, RI, 1993.
  • Z. Zhang, “On degenerate Monge–Ampère equations over closed Kähler manifolds”, Int. Math. Res. Not. 2006 (2006), Art. ID #63640.