By Christoph Hummel
Mikhail Gromov brought pseudo-holomorphic curves into symplectic geometry in 1985. due to the fact that then, pseudo-holomorphic curves have taken on nice value in lots of fields. the purpose of this publication 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 standpoint. houses of specific curiosity are isoperimetric inequalities, a monotonicity formulation, gradient bounds and the elimination of singularities. a different bankruptcy is dedicated to suitable good points of hyperbolic surfaces, the place pairs of pants decomposition and thickthin decomposition are defined. The e-book is largely self-contained and may even be obtainable to scholars with a uncomplicated wisdom of differentiable manifolds and protecting areas.
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Extra info for Gromov’s Compactness Theorem for Pseudo-holomorphic Curves
Example text
Since K is compact, there exists a positive constant C E (0, 1] and a sufficiently small neighbourhood V of Kin [D,M] such that da(v,J(1)v) ~ CJ1(1) (v, v) for each v E TV c T[D,M]. Moreover, a is bounded by some constant C(a) on V. Hence the map j For any compact J-holomorphic curve f: S ~ M with boundary and connected domain S satisfying d(f) < En' the isoperimetric inequality from the previous lemma holds with the same constant cn. II. Estimates for area and first derivatives 30 Proof Assume that f : S ~ M is a compact I-holomorphic curve with boundary and connected domain S. Suppose that the area of f is smaller than C ML 106. After applying the monotonicity lemma as in the last proof it follows that d(f(s),f(dS» $ J~(f) < 100 for any s E S. 3. Isoperimetric inequalities for I-holomorphic maps Namely, let cp: S ~ 23 M be smooth. Then Wirtinger's inequality implies that d(cp) = L(jq>'Jl~ Lcp*w and equality holds if cp is I-holomorphic. , by Stokes' theorem. The claims (i) and (ii) follow. 1) for closed I -holomorphic curves I in M depending only on the homology class of I. 3. Isoperimetric inequalities for J-holomorphic maps Definition. Let fli be a family of maps from surfaces with boundary to a fixed Riemannian manifold (M, g). J(d(f)) for any I E fli.