1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 1. PROBLEMS, BASIC CONCEPTS AND OVERVIEW 47 Given J and (x, T) this interval only depends on the homotopy class [u] of u ...

48 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS We will not prove this result in this paper. Nevertheless it is useful to under- st ...

LECTURE 1. PROBLEMS, BASIC CONCEPTS AND OVERVIEW 49 Theorem 1.37. An area preserving self map of the open annulus having one per ...

50 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS This prompts a certain number of conjectures. Conjecture 1.41. On a homotopy three ...

LECTURE 1. PROBLEMS, BASIC CONCEPTS AND OVERVIEW 51 there is still something like a system of surfaces of section. We will only ...

52 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS together with t he cylinder (C \ {O}) x {O} under iI> give a finite energy folia ...

LECTURE 1. PROBLEMS, BASIC CONCEPTS AND OVERVIEW 53 By μ( z ), z E r , we denote, of course, the Conley-Zehnder index of the cor ...

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LECTURE 2 Analytical Tools In this chapter we introduce some of the analysis necessary for the applications given later. A compr ...

56 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS Proof. Using a finite covering argument it suffices to show that for f.^1 > 0 sm ...

LECTURE 2. ANALYTICAL TOOLS 57 Study the Sobolev embedding results, which are related to the W^1 ·P-norms p > 2. Assume nex ...

58 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS 2.2. Bubbling-off analysis Assume we are given a sequence iie: D ___, IR x M of smo ...

LECTURE 2. ANALYTICAL TOOLS 59 The third assertion gives a characterisation of completeness which is a version of Ekeland's vari ...

60 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS The first observation is the following Proposition 2.8. Let u be as in (6). Then (7 ...

Moreover, LECTURE 2. ANALYTICAL TOOLS E(vk I D1) 2 ~ E(u) 8svk + l8tvk = 0 on D1 2 JD, vk_d>.--tO. 2 61 Now applying Theorem ...

62 and H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS Vs+ lvt = 0 E(v) < oo { v(O,-)*>-=m Js, { v*d>-=O. lbs' We leave it as ...

LECTURE 2. ANALYTICAL TOOLS 63 In the previous theorem we showed that every sequence sk ---> oo has a subse- quence so that l ...

64 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS In order to describe this in more detail assume next we have a finite energy half c ...

LECTURE 2. ANALYTICAL TOOLS 65 for alls?: 0, EE R Moreover, z(s, t) = ef;, r(T)dT[e(t) + r(s, t)] where z = (x, y) and e =/=- 0 ...

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