Let $(M, J, g)$ be a compact almost complex manifold with a Riemannian metric $g$ that preserves the almost complex structure $J$. I want to prove that a holomorphic disk $u: D^2\to M$ of a small area $Area(u)$ can't have a big diameter. Is there a way to give a bound on the diameter of $u(D^2)$ in terms of the area $Area(u)$?
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1$\begingroup$ I'm not an expert on this, but could the monotonicity formula help? Cf. mathoverflow.net/questions/107468/… $\endgroup$– Leo MoosCommented Apr 18, 2021 at 13:50
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1$\begingroup$ This is certainly not true. Even when $M$ is the ordinary Riemann sphere, with standard holomorphic structure. You can have a conformal map $D^2\to S$ whose image has arbitrarily small area, and fixed diameter. $\endgroup$– Alexandre EremenkoCommented Apr 18, 2021 at 15:27
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$\begingroup$ @LeoMoos monotonicity formula will not apply in case the disk u has a thin but long image (big diameter), so the image of u cant be enclosed in a small ball even if u have a very small area. As Alexander pointed out such disks u exist in S^2, thanks LeoMoos, and Alexander for your helpful comments ....... $\endgroup$– Shah FaisalCommented Apr 19, 2021 at 14:23
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