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The degenerate J-flow and the Mabuchi energy on minimal surfaces of general type

Data publikacji: 05.06.2012

Universitatis Iagellonicae Acta Mathematica, 2012, Tom 50, s. 89-106

https://doi.org/10.4467/20843828AM.12.003.1125

Autorzy

,
Jian Song
Rutgers University, Piscataway, NJ, USA
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Ben Weinkove
Northwestern University
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Tytuły

The degenerate J-flow and the Mabuchi energy on minimal surfaces of general type

Abstrakt

We prove existence, uniqueness and convergence of solutions of the degenerate J- ow on Kahler surfaces. As an application, we establish the properness of the Mabuchi energy for Kahler classes in a certain subcone of the Kahler cone on minimal surfaces of general type.

Bibliografia

Pobierz bibliografię

1. Błocki Z., Weak solutions to the complex Hessian equation, Ann. Inst. Fourier (Grenoble), 55, No. 5 (2005), 1735–1756.

2. Chen X., On the lower bound of the Mabuchi energy and its application, Int. Math. Res. Notices, 12 (2000), 607–623.

3. Chen X.,  A  new parabolic flow in  K¨ahler  manifolds, Comm. Anal.  Geom., 12, No. 4 (2004), 837–852.

4. Cherrier P., E´quations de Monge–Amp`ere sur les vari´et´es hermitiennes compactes, Bull. Sci. Math. (2), 111, No. 4 (1987), 343–385.

5. Chinh  L. H.,  Solutions  to  degenerate complex Hessian equations,  preprint,   arXiv: 1202.2436.

6. Dinew S., Ko-lodziej S., Liouville  and Calabi–Yau type theorems for  complex Hessian equations, preprint,  arXiv.org/1203.3995.

7. Donaldson S. K., Moment maps and diffeomorphisms, Asian J. Math.,  3, No. 1 (1999), 1–16.

8. Donaldson S. K., Scalar curvature and stability of toric  varieties, J. Differential  Geom., 62 (2002), 289–349.

9. Fang H., Lai M., On the geometric flows solving K¨ahlerian inverse σk  equations, Pacific J. Math., 258, No. 2 (2012), 291–304.

10. Fang H., Lai M., Convergence of general inverse σk -flow on K¨ahler manifolds with Calabi Ansatz, preprint,  arxiv:1203.5253.

11. Fang H., Lai M., Ma X., On a class of fully nonlinear flows in K¨ahler geomety, J. Reine Angew. Math., 653 (2011), 189–220.

12. Fang H., Lai M., Song J., Weinkove B.,  The J -flow on K¨ahler surfaces: a boundary case,  preprint,  arXiv:1204.4068.

13. Guan B., Li Q., A Monge–Ampere Type Fully Nonlinear Equation on Hermitian  Mani- folds,  Discrete Contin. Dyn. S., series B, 17, No. 6 (2012), 1991–1999.

14. Hou Z., Complex Hessian equation on K¨ahler manifold, Int. Math. Res. Not. IMRN 2009, No. 16,  3098–3111.

15. Hou Z., Ma X. N., Wu D., A second order estimate for complex Hessian equations on a compact  K¨ahler manifold, Math. Res. Lett.,  17, No. 3 (2010), 547–561.

16. Kołodziej S., The complex Monge–Amp`ere  equation, Acta  Math.,  180,  No. 1 (1998), 69–117.

17. Kołodziej S., The Monge–Amp`ere equation on compact K¨ahler manifolds, Indiana Univ. Math. J., 52, No. 3 (2003), 667–686.

18. Kołodziej S., The  complex Monge–Amp`ere  equation and pluripotential  theory,  Mem. Amer. Math. Soc., 178, No. 840 (2005).

19. Li S.-Y., On the Dirichlet  problems for symmetric function equations of the eigenvalues of the  complex Hessian, Asian J. Math., 8, No. 1 (2004), 87–106

20. Li  Y.,  A  priori  estimates on Donaldson equation over compact Hermitian  manifolds,  preprint,  arXiv:   1210.0254.

21. Phong D. H.,  Song J.,  Sturm  J.,  Weinkove  B.,  The  Moser–Trudinger  inequality  on K¨ahler–Einstein manifolds, Amer. J. Math., 130, No. 4 (2008), 1067–1085.

22. Phong D. H., Sturm  J., The Dirichlet  problem for degenerate complex Monge–Amp`ere equations,  Comm. Anal. Geom., 18, No. 1 (2010), 145–170.

23. Panov D., Ross J., Slope stability  and exceptional divisors of high genus, Math.  Ann., 343, No. 1 (2009), 79–101.

24. Ross J., Thomas R., An obstruction to the existence of constant scalar curvature K¨ahler  metrics, J. Differential  Geom., 72, No. 3 (2006), 429–466.

25. Song J., Tian G., The K¨ahler–Ricci flow through singularities, preprint, arXiv:0909.4898 [math.DG].

26. Song J., Weinkove B.,  The convergence and singularities of the J-flow with applications to the  Mabuchi energy, Comm. Pure Appl. Math., 61, No. 2 (2008), 210–229.

27. Song J., Weinkove B., Contracting  exceptional divisors by the K¨ahler–Ricci  flow, Duke Math. J. 162, No. 2 (2013), 367–415.

28. Tian G., K¨ahler–Einstein metrics with positive scalar curvature, Invent. Math., 130, No. 1 (1997), 1–37.

29. Tian G., Canonical metrics in K¨ahler geometry. Notes taken by Meike Akveld. Lectures in  Mathematics ETH Zu¨rich. Birkh¨auser Verlag, Basel, 2000.

30. Tosatti  V.,  Weinkove  B.,  On the evolution of a Hermitian  metric  by its Chern–Ricci form,  preprint,  arXiv:1201.0312.

31. Tsuji  H., Existence and degeneration of K¨ahler–Einstein  metrics  on minimal  algebraic  varieties of general type, Math. Ann., 281 (1988), 123–133.

32. Weinkove B., Convergence of the J -flow on K¨ahler  surfaces, Comm. Anal. Geom., 12, No. 4  (2004), 949–965.

33. Weinkove  B.,  On the J -flow  in  higher dimensions and the lower boundedness  of the Mabuchi energy, J. Differential  Geom., 73, No. 2 (2006), 351–358.

34. Yau S.-T.,  On the Ricci curvature of a compact K¨ahler manifold and the complex Monge– Amp`ere  equation, I, Comm. Pure Appl. Math., 31 (1978), 339–411.

35. Yau S.-T., Open problems in geometry, Proc. Symposia Pure Math., 54 (1993), 1–28.

Informacje

Informacje: Universitatis Iagellonicae Acta Mathematica, 2012, Tom 50, s. 89-106

Typ artykułu: Oryginalny artykuł naukowy

Autorzy

Rutgers University, Piscataway, NJ, USA

Northwestern University

Publikacja: 05.06.2012

Status artykułu: Otwarte __T_UNLOCK

Licencja: Żadna

Udział procentowy autorów:

Jian Song (Autor) - 50%
Ben Weinkove (Autor) - 50%

Korekty artykułu:

-

Języki publikacji:

Angielski

The degenerate J-flow and the Mabuchi energy on minimal surfaces of general type

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