Download Gromov’s Compactness Theorem for Pseudo-holomorphic Curves by Christoph Hummel PDF
By Christoph Hummel
Mikhail Gromov brought pseudo-holomorphic curves into symplectic geometry in 1985. when you consider 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 houses of pseudo-holomorphic curves are investigated and proved from a geometrical perspective. homes 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 ebook is basically self-contained and will even be available to scholars with a simple wisdom of differentiable manifolds and masking areas.
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The area of f is finite. As usual we denote by of C. ::,oo IZnl » = 0, {z E C Ilzl < r } III. sI1(fID \{O}) ro = 1'JD( ro \{o} E (0, 1) such that e(o(fio » ~ £ for each r IITfl12 pdpd8 ~ I:0 (2np)-1 (fo21t IITfl1 Pd8) 2dp = ~ (0 e2 (0(flo) 2nJo p dp ~ £2 (0 2nJo dp p = 00 . Since feD \ {O}) is relatively compact in M, we may suppose that (f(zn» converges in M, say lim f(zn) =: P EM. n-,>oo Assume there is another sequence (Wn)n~l in D \ {O} with lim wn = 0 and limf(w n) = IZnl and sn := Iwnl we may assume without loss of generality that q 1- p.
Ii) There exists a neighbourhood G of a in S and an open neighbourhood U of f(G \ {a}) in M such that the following holds. There exists a bounded i-form {3 on U satisfying d{3(v,]v);;:: /leV, v)for each v E TU c TM. Then f can be extended to a 1 -holomorphic map on S. Observe that the statement is purely local. So it is sufficient to prove the theorem for S = G = D the open unit disc in C and a = O. 1 with S = D = G and a = O. The proof has essentially two steps, namely: (a) There is a continuous extension of f to D.
11) that va JaM ~ L, ( J a(cy} - J a(b y ) ) ~ K' L,(c y - y and this finishes the proof of the monotonicity lemma. 3. 11), one could argue more directly in the last proof. J: 3. 1 can even be sharpened. 4. 1 and of the monotonicity lemma are satisfied. Let f: S ~ M be a compact J -holomorphic curve with boundary and connected domain S. Assume that its diameter 8(f) := sup {d(f(s), f(s')) I s, s' E S } is smaller than EO' Then f satisfies the isoperimetric inequality for some constant cn > 0 depending only on (M, J, 11), EO and the number n ofboundary components of S.