By Christoph Hummel

Mikhail Gromov brought pseudo-holomorphic curves into symplectic geometry in 1985. because then, pseudo-holomorphic curves have taken on nice significance in lots of fields. the purpose of this ebook is to provide the unique evidence of Gromov's compactness theorem for pseudo-holomorphic curves intimately. neighborhood homes of pseudo-holomorphic curves are investigated and proved from a geometrical perspective. houses of specific curiosity are isoperimetric inequalities, a monotonicity formulation, gradient bounds and the elimination of singularities. a different bankruptcy is dedicated to suitable beneficial properties of hyperbolic surfaces, the place pairs of pants decomposition and thickthin decomposition are defined. The publication is basically self-contained and may even be obtainable to scholars with a simple wisdom of differentiable manifolds and masking areas.

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**Sample text**

D Chapter IV Hyperbolic surfaces This chapter describes the pairs of pants decomposition and the thick-thin decomposition of oriented complete hyperbolic surfaces of finite area. Using the pairs of pants decomposition, one gets, roughly speaking, a parametrization of the space of hyperbolic structures on such a surface which coincides with the space of its complex structures. The thick-thin decomposition gives a classification of the thin parts of a hyperbolic surface, which are the components of small injectivity radius.

Area minimizing maps from a compact surface with boundary to some Euclidean space satisfy the well-known isoperimetric inequality 4n . area ~ (length of boundary)2 (see Appendix A) which is "similar" to the one claimed in the lemma. II. 1. Let R = injrad(M, Il) be the injectivity radius of M. 1). Let expp := expp IBR(O) denote the exponential map of Mat p restricted to the R-ball BR(O) c (TpM,Jp,ll p ) where p is an arbitrary point in M fixed from now on. We identify canonically the tangent spaces to TpM with TpM itself.

2. Chapter III Higher order derivatives This chapter is concerned with the estimate of higher order derivatives of f-holomorphic maps. For that purpose the differentials of f -holomorphic maps are made into pseudo-holomorphic maps to which the Gromov-Schwarz lemma applies. This is used in the proof of Gromov's theorem on the removal of singularities for f-holomorphic maps. It is a generalization of a theorem of Riemann from complex analysis, which says that a holomorphic map f: S \ {a} ~ S2 from a Riemann surface minus an interior point a to the Riemann sphere can be extended to a holomorphic map S ~ S2, provided f does not have an essential singularity at a.