Geodesic
Holomorphic Variations of Minimal Disks with Boundary on a Lagrangian Surface
Canadian journal of mathematics, Tome 62 (2010) no. 6, pp. 1264-1275

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DOI

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.
DOI : 10.4153/CJM-2010-068-1
Mots-clés : 58E12, 53C21, 53C26 
Chen, Jingyi; Fraser, Ailana. Holomorphic Variations of Minimal Disks with Boundary on a Lagrangian Surface. Canadian journal of mathematics, Tome 62 (2010) no. 6, pp. 1264-1275. doi: 10.4153/CJM-2010-068-1
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     title = {Holomorphic {Variations} of {Minimal} {Disks} with {Boundary} on a {Lagrangian} {Surface}},
     journal = {Canadian journal of mathematics},
     pages = {1264--1275},
     year = {2010},
     volume = {62},
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     doi = {10.4153/CJM-2010-068-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2010-068-1/}
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