Get An Introduction to Compactness Results in Symplectic Field PDF

By Casim Abbas

ISBN-10: 3642315429

ISBN-13: 9783642315428

ISBN-10: 3642315437

ISBN-13: 9783642315435

This publication presents an advent to symplectic box conception, a brand new and critical topic that's presently being built. the start line of this conception are compactness effects for holomorphic curves proven within the final decade. the writer offers a scientific creation offering loads of heritage fabric, a lot of that is scattered through the literature. because the content material grew out of lectures given by way of the writer, the most goal is to supply an access element into symplectic box idea for non-specialists and for graduate scholars. Extensions of convinced compactness effects, that are believed to be precise by means of the experts yet haven't but been released within the literature intimately, refill the scope of this monograph.

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Additional info for An Introduction to Compactness Results in Symplectic Field Theory

Example text

As above this shows that our initial assumption β(x1 ) < 0 cannot hold, so that β(z) ≥ 0 for all z ∈ S. Summarizing, we get β ≡ 0 and λ ≡ 1 so that the two metrics are indeed equal as claimed. As a consequence, if two metrics with constant sectional curvature −1 on a compact surface induce the same complex structure they must be equal since they are conformal. Let j be a complex structure on S. 26). Therefore the hyperbolic metric gH + on H descends to a metric h on the quotient, and the covering projection (H + , gH + ) → (S, h) becomes a local isometry.

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 { k } of the boundaries of the pants and the twist parameters {αj } ⊂ [0, 1] used to glue them together. Since we are now considering isometric stable surfaces equivalent, we may assume that 0 ≤ α ≤ 1. If we now have a sequence (S, jn ) of such surfaces we would like to define a notion of convergence (n) (n) based on the data {{ k }, {αj }}.

This time the maximum principle prohibits β from having an interior minimum on B unless it is constant. As above this shows that our initial assumption β(x1 ) < 0 cannot hold, so that β(z) ≥ 0 for all z ∈ S. Summarizing, we get β ≡ 0 and λ ≡ 1 so that the two metrics are indeed equal as claimed. As a consequence, if two metrics with constant sectional curvature −1 on a compact surface induce the same complex structure they must be equal since they are conformal. Let j be a complex structure on S.

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


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