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# Holomorphic variations of minimal disks with boundary on a Lagrangian surface

Published:2010-08-18
Printed: Dec 2010
• Jingyi Chen,
Department of Mathematics, The University of British Columbia, Vancouver, BC, V6T 1Z2
• Ailana Fraser,
Department of Mathematics, The University of British Columbia, Vancouver, BC, V6T 1Z2
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## Abstract

Let $L$ be an oriented Lagrangian submanifold in an $n$-dimensional Kähler manifold~$M$. Let $u \colon D \to M$ 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.
 MSC Classifications: 58E12 - Applications to minimal surfaces (problems in two independent variables) [See also 49Q05] 53C21 - Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] 53C26 - Hyper-Kahler and quaternionic Kahler geometry, special'' geometry

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