Collapsing of Calabi-Yau manifolds and special Lagrangian submanifolds

Yuguang Zhang


In this paper, the relationship  between the existence of special Lagrangian submanifolds and the collapsing of Calabi–Yau manifolds is studied. First, special Lagrangian fibrations are constructed on some regions of bounded curvature and sufficiently collapsed in Ricci-flat Calabi–Yau manifolds. Then, conversely, it is shown that  the existence of special Lagrangian submanifolds with small volume implies the collapsing of some regions in the ambient Calabi–Yau manifolds.


1. Anderson M. T.,  Ricci  curvature  bounds and Einstein  metrics  on compact manifolds, J. Amer. Math. Soc., 2 (1989), 455–490.

2. Anderson M. T.,  Convergence and rigidity  of manifolds under Ricci  curvature bounds, Invent.  Math., 102 (1990), 429–445.

3. Anderson M. T., The L2   structure of moduli spaces of Einstein metrics on 4-manifolds, G.A.F.A.,  (1991), 231–251.

4. Cheeger J., Degeneration of Einstein metrics and metrics with special holonomy, in Sur- veys in  differential  geometry VIII, 29–73.

5. Cheeger J., Colding T. H., On the structure of space with Ricci curvature bounded below I, Jour.  of Diff. Geom., 46 (1997), 406–480.

6. Cheeger J., Colding T. H., On the structure of space with Ricci curvature bounded below II,  Jour. of Diff. Geom., 52 (1999), 13–35.

7. Cheeger J., Colding T. H., Tian  G., On the singularities  of spaces  with bounded Ricci  curvature, Geom. Funct. Anal., 12 (2002), 873–914.

8. Cheeger J., Fukaya K.,  Gromov M., Nilpotent structures and invariant  metrics on collapsed manifolds, Joural of the American Mathematical Society, 5 (1992), 327–372.

9. Cheeger J., Gromov M., Collapsing Riemannian manifolds while keeping their curvature bound I, J.  Differ. Geom., 23 (1986), 309–364.

10. Cheeger J., Gromov M., Collapsing Riemannian manifolds while keeping their curvature bounded  II, J. Diff. Geom., 32 (1990), 269–298.

11. Cheeger J., Tian G., Anti-self-duality  of curvature and degeneration of metrics with spe-cial  holonomy, Commun. Math. Phys., 255 (2005), 391–417.

12. Cheeger J., Tian G., Curvature and injectivity  radius estimates for Einstein 4-manifolds,  Journal of the American Mathematical Society, 19 (2006), 487–525.

13. Duistermaat  J.,  On global action-angle coordinates, Comm. Pure Appled  Math.,  33 (1980), 687–706.

14. Fukaya K., Hausdorff convergence of Riemannian manifolds and its application, Advance Studies in Pure Mathematics, 18 (1990), 143–234.

15. Fukaya K., Multivalued Morse theory, asymptortic analysis and mirror aymmetry, Pro- ceedings of  Symposia in Pure Mathematics, 73 (2005), 205–278.

16. Green R. E., Wu H., Lipschitz converges of Riemannian manifolds, Pacific J. Math., 131 (1988), 119–141.

17. Gromov M., Metric  structures for Riemannian and non-Riemannian spaces, Birkh¨auser 1999.

18. Goldstein E., A  construction  of new families of minimal  lagrangian submanifolds via torus  action, J. Diff. Geom., 58 (2001), 233–261.

19. Goldstein E., Calibrated fibrations, Comm. Anal. Geom., 10 (2002), 127–150.

20. Gross M., Examples of special lagrangian fibrations, in Symplectic Geometry and Mirror Symmetry, World Scientific Singapore, (2001), 81–109.

21. Gross M., Special lagrangian fibrations II-Geometry,  Surveys in Differential  Geometry:  Differential  geometry inspired by string theory, International  Press, (1999), 341–404.

22. Gross M., Wilson P. M. H., Mirror symmetry via 3-tori for a class of Calabi–Yau treefolds,  Math. Ann., 309 (1997), 505–531.

23. Gross M., Wilson P. M. H., Large complex structure limits of K3 surfaces, J. Diff. Geom., 55 (2000), 475–546.

24. Gilbarg D., Trudinger N. S., Elliptic  partial differential equations of second two, Springer 1983.

25. Gukov S., Yau S. T., Zaslow E., Duality and fibrations on G2  manifolds, Turkish Journal of  Mathematics, 27 (2003), 61–97.

26. Harvey R., Lawson H. B., Calibrated geometries, Acta Math., 148 (1982), 47–157. 

27. Hitchin N., The moduli space of special lagrangian submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 503–515.

28. Joyce D. D., Compact manifolds with special holonomy, Oxford University  Press, 2000.

29. Joyce D. D., Singularities of special lagrangian fibrations and the SYZ conjecture, Comm. Anal. Geom., 11 (2003), 859–907.

30. Joyce D. D., Lectures on Calabi–Yau and special Lagrangian geometry, math.DG/0108088. 

31. Kontsevich  M.,  Soibelman Y.,  Homological mirror symmetry and torus fibrations,  in Symplectic geometry and mirror symmetry, World Sci. Publishing, (2001), 203–263.

32. Lu P., K¨ahler–Einstein metrics on Kummer threefold and special lagrangian tori, Comm. Anal. Geom., 7 (1999), 787–806.

33. Mclean R. C., Deformation  of calibrated submanifolds, Comm. Anal. Geom., 6 (1998), 705–747.

34. Petersen P., Riemannian Geometry, Springer, 1997.

35. Ruan W. D., On the convergence and collapsing of K¨ahler metrics, J. Differ. Geom., 52 (1999), 1–40.

36. Ruan W. D., Generalized special Lagrangian torus fibration for Calabi–Yau hypersurfaces in  toric varieties. I. Commun. Contemp. Math., 9 (2007), no. 2, 201–216.

37. Ruan W. D., Generalized special Lagrangian torus fibrations for Calabi–Yau hypersurfaces in  toric  varieties. II. in Mirror  symmetry. V, Amer. Math.  Soc., Providence, RI, 2006. 457–477.

38. Ruan W. D., Generalized special Lagrangian fibration  for  Calabi–Yau hypersurfaces in toric varieties III: The smooth fibres, arXiv:math/0309450.

39. Ruan W. D., Zhang Y., Convergence of Calabi–Yau manifolds, Advances in Mathematics Volume 228, Issue 3, (2011), 1543–1589.

40. Salur S., Deformations of special lagrangian submanifolds, Comm. Cont. Math.,  Vol. 2, 3 (2000), 365–372.

41. Strominger A., Yau S. T., Zaslow E., Mirror symmetry is T-duality,  Nucl. Phys. B, 479 (1996), 243–259.

42. Thomas R. P., Yau S. T., Special Lagrangians, stable bundles and mean curvature flow, Comm.  Anal. Geom., 10 (2002), no. 5, 1075–1113.

43. Tosatti  V.,  Limits  of Calabi–Yau metrics when the K¨ahler  class degenerates,  J. Eur. Math. Soc., 11 (2009), no. 4, (2009), 755–776.

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

45. Yau S. T.,  Einstein manifolds with zero Ricci  curvature, in Lectures on Einstein mani- folds,  International  Press, (1999), 1–14.

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