Download From Stein to Weinstein and Back: Symplectic Geometry of by Kai Cieliebak, Yakov Eliashberg PDF
By Kai Cieliebak, Yakov Eliashberg
This publication is dedicated to the interaction among advanced and symplectic geometry in affine advanced manifolds. Affine advanced (a.k.a. Stein) manifolds have canonically outfitted into them symplectic geometry that's chargeable for many phenomena in complicated geometry and research. The target of the booklet is the exploration of this symplectic geometry (the street from "Stein to Weinstein") and its functions within the complicated geometric international of Stein manifolds (the street "back"). this can be the 1st booklet which systematically explores this connection, hence delivering a brand new method of the classical topic of Stein manifolds. It additionally comprises the 1st exact research of Weinstein manifolds, the symplectic opposite numbers of Stein manifolds, which play a major position in symplectic and make contact with topology. Assuming just a basic heritage from differential topology, the e-book presents introductions to a few of the concepts from the speculation of services of a number of advanced variables, symplectic geometry, h-principles, and Morse thought that input the proofs of the most effects. the most result of the e-book are unique result of the authors, and a number of other of those effects seem the following for the 1st time. The booklet might be priceless for all scholars and mathematicians attracted to geometric elements of advanced research, symplectic and make contact with topology, and the interconnections among those topics.
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Additional resources for From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds
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SMOOTHING and denote them by the same letters. Consider the standard complex coordinates zj = xj + iyj on Cn and write z = (u, v, w) with u := (x1 , . . , x ), v = (x +1 , . .
14. Let Σ be a cooriented hypersurface in a K¨ ahler manifold (X, J, ω) with second fundamental form IIΣ . 14) LΣ (X) = IIΣ (X) + IIΣ (JX). In particular, Σ is J-convex if and only if at every point x ∈ Σ the mean normal curvature along any complex line in Tx Σ is positive. 7. Examples of J-convex functions and hypersurfaces An important class of hypersurfaces are boundaries of tubular neighborhoods of submanifolds. In this section we examine their J-convexity for the cases of totally real submanifolds and complex hypersurfaces.
Since W ∩ JW = {0}, W contains a complex line L. Let C be a complex curve through p tangent to L. Then φ|C attains a local maximum at p. 2. 2, we can speak about continuous J-convex functions on almost complex manifolds as functions whose restrictions to all complex curves are subharmonic. Such functions are also called (strictly) plurisubharmonic. 5. A continuous function φ : Cn ⊃ U → R is i-convex if and only if its restriction to every complex line is subharmonic. This means that there exists a positive continuous function m : U → R such that 2π 1 1 φ(z + weiθ )dθ φ(z) + m(z)|w|2 ≤ 4 2π 0 for all z ∈ U and sufficiently small w ∈ Cn (depending on z).