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Holomorphic Variations of Minimal Disks with Boundary on a Lagrangian Surface

  • Jingyi Chen (a1) and Ailana Fraser (a1)
Abstract

Let L be an oriented Lagrangian submanifold in an n-dimensional Kähler manifold M. Let u: DM be a minimal immersion from a disk D with u(𝜕D) ⊂ L such that u(D) meets L orthogonally along u(𝜕D). Then the real dimension of the space of admissible holomorphic variations is at least n + μ(E, F), where μ(E, F) is a boundary Maslov index; the minimal disk is holomorphic if there exist n admissible holomorphic variations that are linearly independent over ℝ at some point p ∈ 𝜕D; if M = ℂPn and u intersects L positively, then u is holomorphic if it is stable, and its Morse index is at least n + μ(E, F) if u is unstable.

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Footnotes
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This work is partially supported by NSERC.

Footnotes
References
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[1] Arezzo, Claudio, Stable complete minimal surfaces in hyper-Kähler manifolds. Compositio Math. 112(1998), 33–40. doi:10.1023/A:1000358906964
[2] Fraser, Ailana, On the free boundary variational problem for minimal disks. Comm. Pure Appl. Math. 53(2000), 931–971. doi:10.1002/1097-0312(200008)53:8h931::AID-CPA1i3.0.CO;2-9
[3] Mc Duff, Dusa and Salamon, Dietmar, J-holomorphic curves and symplectic topology. American Mathematical Society Colloquium Publications 52, American Mathematical Society, Providence, RI, 2004.
[4] Micallef, Mario, Stable minimal surfaces in Euclidean space. J. Differential Geom. 19(1984), 57–84.
[5] Micallef, Mario and Douglas Moore, John, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes. Ann. of Math. (2) 127(1988), 199–227. doi:10.2307/1971420
[6] Micallef, Mario and Wang, Mc Kenzie, Metrics with nonnegative isotropic curvature. Duke Math. J. 72(1993), 649–672. doi:10.1215/S0012-7094-93-07224-9
[7] Siu, Yum-Tong and Yau, Shing-Tung, Compact Kähler manifolds of positive bisectional curvature. Invent. Math. 59(1980), 189–204. doi:10.1007/BF01390043
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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