1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

Lecture 3. The Weinstein Conjecture in the Overtwisted Case Up to this point we described some of the results which we would lik ...

68 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS • The distribution T(V \ { e}) n ~, which is one-dimensional defines a foliation ha ...

LECTURE 3. THE WEINSTEIN CONJECTURE IN THE OVERTWISTED CASE 69 by the set 1 {(x, y, -2xy) I (x, y) E B 0 (0) C JR^2 } with some ...

70 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS J(x,y,z) ( Jx ) - ( ~1 ) and we continue J arbitrarily onto the whole manifold. In ...

LECTURE 3. THE WEINSTEIN CONJECTURE IN THE OVERTWISTED CASE 71 and J(z,O)=i if lzl~l where F := {(z,azkl^2 ) E C^2 I z E 8D; a E ...

72 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS Proposition 3.4. Let 2 < p < oo and U be a sufficiently small neighbourhood o ...

LECTURE 3. THE WEINSTEIN CONJECTURE IN THE OVERTWISTED CASE 73 we have C^2 = X z EB T(z,o)F and we see that the Banach space V' ...

74 Then H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS D81(uo)(h, k) T(h, k) (hs, ks) + J( uo) (ht, kt) + [DJ( uo)(h, k )] · ( i, 0) ...

LECTURE 3. THE WEINSTEIN CONJECTURE IN THE OVERTWISTED CASE 75 Proof. Consider first elements of the form (h, 0) in ker(T). Thes ...

76 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS The linearisation T of the Cauchy Riemann operator is given by T(h, k) + (hs + iht ...

LECTURE 3. THE WEINSTEIN CONJECTURE IN THE OVERTWISTED CASE 77 Now we have 0 = dd I - 7 H(T, a(T)) t =O = DH(0,0)(1, a'(O)) = D8 ...

78 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS Then J(O, z) = z and therefore D 2 f(O, z ) =Id for all z E D. We find now by the i ...

LECTURE 3. THE WEINSTEIN CONJECTURE IN THE OVERTWISTED CASE 79 3.3. A short summary and the prolongation of the Bishop family Le ...

80 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS geometrically distinct solutions u,,, where <I> : ( T - {j' T + 8) x D --+ JR ...

LECTURE 3. THE WEINSTEIN CONJECTURE IN THE OVERTWISTED CASE 81 Proposition 3.15. There exists an 0 < c < 1 and a constant ...

82 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS In addition, there is a sequence zk E D+, Zk ---+ 0 and ck ---+ 0, ck > 0 satisf ...

LECTURE 3. THE WEINSTEIN CONJECTURE IN THE OVERTWISTED CASE 83 Lemma 3.16. Let w: D \ { 1} ---+ IR x M be a smooth map satisfyin ...

84 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS we can make the choices in such a way that the length of uk(ri) is smaller than c f ...

LECTURE 3. THE WEINSTEIN CONJECTURE IN THE OVERTWISTED CASE 85 along the leaves vanish. On the other hand, by Stokes, (30) where ...

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