Abstract view
Holomorphic variations of minimal disks with boundary on a Lagrangian surface


Published:20100818
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
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.