Abstract
The goal of this chapter is to prove Gromov’s monotonicity lemma and his Schwarz lemma for J-holomorphic curves using isoperimetric inequalities. The Gromov-Schwarz lemma is a generalization of the classical Schwarz lemma from complex analysis which states that for any holomorphic map f from the open unit disc in ℂ into itself with f (0)= 0 its derivative at 0 is bounded from above by one. For any compact J-holomorphic curve f : S → (M, J) in a compact almost complex manifold the area of a piece of f (S), cut from f (S) by a small r-ball in M centred on the image f (S), is estimated from below in terms of r by the monotonicity lemma.
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© 1997 Springer Basel AG
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Hummel, C. (1997). Estimates for area and first derivatives. In: Gromov’s Compactness Theorem for Pseudo-holomorphic Curves. Progress in Mathematics, vol 151. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8952-0_3
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DOI: https://doi.org/10.1007/978-3-0348-8952-0_3
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9842-3
Online ISBN: 978-3-0348-8952-0
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