Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-19T03:26:49.198Z Has data issue: false hasContentIssue false

Holomorphic Variations of Minimal Disks with Boundary on a Lagrangian Surface

Published online by Cambridge University Press:  20 November 2018

Jingyi Chen*
Affiliation:
Department of Mathematics, The University of British Columbia, Vancouver, BC, V6T 1Z2
Ailana Fraser*
Affiliation:
Department of Mathematics, The University of British Columbia, Vancouver, BC, V6T 1Z2
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2010

Footnotes

This work is partially supported by NSERC.

References

[1] Arezzo, Claudio, Stable complete minimal surfaces in hyper-Kähler manifolds. Compositio Math. 112(1998), 33–40. doi:10.1023/A:1000358906964Google Scholar
[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;2-9Google Scholar
[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.Google Scholar
[4] Micallef, Mario, Stable minimal surfaces in Euclidean space. J. Differential Geom. 19(1984), 57–84.Google Scholar
[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:10.2307/1971420Google Scholar
[6] Micallef, Mario and Wang, Mc Kenzie, Metrics with nonnegative isotropic curvature. Duke Math. J. 72(1993), 649–672. doi:10.1215/S0012-7094-93-07224-9Google Scholar
[7] Siu, Yum-Tong and Yau, Shing-Tung, Compact Kähler manifolds of positive bisectional curvature. Invent. Math. 59(1980), 189–204. doi:10.1007/BF01390043Google Scholar