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 complicated and symplectic geometry in affine complicated manifolds. Affine complicated (a.k.a. Stein) manifolds have canonically outfitted into them symplectic geometry that is liable for many phenomena in advanced geometry and research. The aim of the publication is the exploration of this symplectic geometry (the highway from "Stein to Weinstein") and its functions within the complicated geometric global of Stein manifolds (the street "back"). this is often the 1st e-book which systematically explores this connection, hence offering a brand new method of the classical topic of Stein manifolds. It additionally comprises the 1st certain research of Weinstein manifolds, the symplectic opposite numbers of Stein manifolds, which play a big position in symplectic and phone topology. Assuming just a basic historical past from differential topology, the booklet presents introductions to many of the suggestions from the idea of services of numerous complicated variables, symplectic geometry, h-principles, and Morse thought that input the proofs of the most effects. the most result of the ebook are unique result of the authors, and a number of other of those effects look right here for the 1st time. The ebook might be priceless for all scholars and mathematicians attracted to geometric elements of complicated research, symplectic and speak to topology, and the interconnections among those topics.
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Extra resources for From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds
Example text
From families of hypersurfaces to J-convex functions The following result shows that a continuous family of J-convex hypersurfaces transverse to the same vector field gives rise to a smooth function with regular J-convex level sets. This will be extremely useful for the construction of J-convex functions with prescribed critical points. 4. 25. Let (M ×[0, 1], J) be a compact complex manifold such that M × {0} and M × {1} are J-convex cooriented by ∂r , where r is the coordinate on [0, 1]. Suppose there exists a smooth family (Σλ )λ∈[0,1] of J-convex hypersurfaces transverse to ∂r with Σ0 = M × {0} and Σ1 = M × {1}.
Then ∇ϑ ϑ = µ∇φ φ on L for a positive function µ : L → R+ (f ) Assume in addition that r∂r φ ≥ µr2 and r∂r ψ ≥ µr2 , where r is the distance from L with respect to some Hermitian metric and µ > 0 a constant. Then we can arrange that r∂r ϑ ≥ µr2 /2. 27. 26, φ and ϑ can be connected by the family of J-convex functions φt := (1−t)φ+tϑ, t ∈ [0, 1], satisfying properties (b-f). (ii) If L is Lagrangian and ∇φ φ and ∇ψ ψ are tangent to L, then so is ∇ϑ ϑ. This follows from the observation that tangency of ∇φ φ to L for L Lagrangian is equivalent to vanishing of dφ◦J on L, which is preserved under convex combinations.
G. 6]), this inequality is equivalent to ∆ψ(z) ≥ 0 in the distributional sense, and therefore to ∆φ(z) ≥ 41 m(z)∆w |w−z|2 = m(z). ψ(z) ≤ Now let (V, J) be an almost complex manifold. A complex curve in V is a 1dimensional complex submanifold of (V, J). Note that the restriction of the almost complex structure J to a complex curve is always integrable. 2. A C 2 -function φ on an almost complex manifold (V, J) is J-convex if and only if its restriction to every complex curve is subharmonic. Proof.