Fix an almost-complex structure $J$ on $\mathbb{R}^{2n}.$ Let $u: (D^2, i) \to (\mathbb{R}^{2n}, J)$ be a $J$-holomorphic disk.
My question: can one prove an a-priori bound on the diameter of $u$ (say in terms of the norm of $J$)?
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4$\begingroup$ What do you mean by 'the diameter'? Are you fixing a metric on $\mathbb{R}^{2n}$? If so, is it assumed compatible with $J$ somehow? In general, I expect the answer is 'no' since, for the standard complex structure and metric on $\mathbb{R}^{2n}$ there is no bound on the diameter of holomorphic discs. $\endgroup$– Robert BryantCommented Apr 17, 2018 at 0:36
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$\begingroup$ @RobertBryant For the setting I had in mind, the curve had boundary in a Lagrangian submanifold. I was hoping to control the "size" of the curve" (with respect to some, say J-compatible, background metric). However, the question as posed is clearly too general, as you pointed out. $\endgroup$– user142700Commented Apr 17, 2018 at 0:50
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1 Answer
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You can Riemann mapping theorem to create a holomorphic disk of any diameter in case $n=2$.
