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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
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Abstract

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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

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