Collapsing of Calabi-Yau manifolds and special Lagrangian submanifolds

Yuguang Zhang

Abstrakt
Abstract. In this paper, the relationship between the existence of special
Lagrangian submanifolds and the collapsing of Calabi{Yau manifolds
is studied. First, special Lagrangian brations are constructed on some regions
of bounded curvature and suciently collapsed in Ricci-
at 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.
References
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 Surveys
in di erential geometry VIII, 29{73.
5. Cheeger J., Colding T. H., On the structure of space with Ricci curvature bounded below I,
Jour. of Di . Geom., 46 (1997), 406{480.
6. Cheeger J., Colding T. H., On the structure of space with Ricci curvature bounded below II,
Jour. of Di . 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.
77
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. Di er. Geom., 23 (1986), 309{364.
10. Cheeger J., Gromov M., Collapsing Riemannian manifolds while keeping their curvature
bounded II, J. Di . Geom., 32 (1990), 269{298.
11. Cheeger J., Tian G., Anti-self-duality of curvature and degeneration of metrics with special
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., Hausdor 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, Proceedings
of Symposia in Pure Mathematics, 73 (2005), 205{278.
16. Green R. E., Wu H., Lipschitz converges of Riemannian manifolds, Paci c J. Math., 131
(1988), 119{141.
17. Gromov M., Metric structures for Riemannian and non-Riemannian spaces, Birkhauser
1999.
18. Goldstein E., A construction of new families of minimal lagrangian submanifolds via
torus action, J. Di . Geom., 58 (2001), 233{261.
19. Goldstein E., Calibrated brations, Comm. Anal. Geom., 10 (2002), 127{150.
20. Gross M., Examples of special lagrangian brations, in Symplectic Geometry and Mirror
Symmetry, World Scienti c Singapore, (2001), 81{109.
21. Gross M., Special lagrangian brations II-Geometry, Surveys in Di erential Geometry:
Di erential 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. Di . Geom.,
55 (2000), 475{546.
24. Gilbarg D., Trudinger N. S., Elliptic partial di erential equations of second two, Springer
1983.
25. Gukov S., Yau S. T., Zaslow E., Duality and brations 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 brations 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 brations, in
Symplectic geometry and mirror symmetry, World Sci. Publishing, (2001), 203{263.
32. Lu P., Kahler{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.
78
34. Petersen P., Riemannian Geometry, Springer, 1997.
35. Ruan W. D., On the convergence and collapsing of Kahler metrics, J. Di er. Geom., 52
(1999), 1{40.
36. Ruan W. D., Generalized special Lagrangian torus bration 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 brations 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 bration for Calabi{Yau hypersurfaces in
toric varieties III: The smooth bres, 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 
ow,
Comm. Anal. Geom., 10 (2002), no. 5, 1075{1113.
43. Tosatti V., Limits of Calabi{Yau metrics when the Kahler class degenerates, J. Eur.
Math. Soc., 11 (2009), no. 4, (2009), 755{776.
44. Yau S. T., On the Ricci curvature of a compact Kahler 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 manifolds,
International Press, (1999), 1{14.

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