Casim Abbas's An Introduction to Compactness Results in Symplectic Field PDF

By Casim Abbas

ISBN-10: 3642315437

ISBN-13: 9783642315435

This ebook offers an advent to symplectic box thought, a brand new and significant topic that's presently being constructed. the place to begin of this thought are compactness effects for holomorphic curves validated within the final decade. the writer offers a scientific creation delivering loads of heritage fabric, a lot of that is scattered during the literature. because the content material grew out of lectures given through the writer, the most objective is to supply an access element into symplectic box concept for non-specialists and for graduate scholars. Extensions of definite compactness effects, that are believed to be real by way of the experts yet haven't but been released within the literature intimately, replenish the scope of this monograph.

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We equip H − with the complex structure −i and the metric y −2 geucl . We construct a surface Y by identifying the points ak (t) with ak′ (t) for k = 1, 2, 3 and 0 ≤ t ≤ 1 (see Fig. 9). The complex structures i on G and −i on G′ fit together. Using Fermi coordinates near the geodesics ak and ak′ which have equal lengths, we see that the surface Y also inherits a hyperbolic metric from the two hexagons G and G′ . 2 Riemann Surfaces and Hyperbolic Geometry 33 Fig. 48 The surface Y together with its complex structure and its hyperbolic metric is called a pair of pants (see Fig.

E. it extends over the punctures. Then we can associate to j a unique hyperbolic metric h with finite area. We now know that S can be decomposed isometrically into 2g − 2 + m + n pairs of pants. The metric h can be recaptured up to diffeomorphism from the lengths {ℓk } of the boundaries of the pants and the twist parameters {αj } ⊂ [0, 1] used to glue them together.

We note that a cusp has finite area since area = [0,1]×[1/2,∞) 1 dx ∧ dy = 2. 42 Show that the projection of δ is the only closed geodesic on Cℓ because every geodesic c in H which contains the points γ (t0 ) and γ ′ (t0 ) for some 30 1 Riemann Surfaces Fig. 7 Standard cusp. The Parabolic cylinder is H/G t0 ∈ R must intersect γ and γ ′ at a different angle, hence c projects to a geodesic on Cℓ which has a self-intersection but is not closed. Similarly, show that there are no closed geodesics on the parabolic cylinder.

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An Introduction to Compactness Results in Symplectic Field Theory by Casim Abbas

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