By Kai Cieliebak, Yakov Eliashberg
This e-book is dedicated to the interaction among advanced and symplectic geometry in affine advanced manifolds. Affine advanced (a.k.a. Stein) manifolds have canonically equipped into them symplectic geometry that is answerable for many phenomena in complicated geometry and research. The objective of the publication is the exploration of this symplectic geometry (the highway from "Stein to Weinstein") and its purposes within the complicated geometric global of Stein manifolds (the highway "back"). this is often the 1st ebook which systematically explores this connection, therefore delivering a brand new method of the classical topic of Stein manifolds. It additionally includes the 1st designated 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 heritage from differential topology, the e-book offers introductions to some of the concepts from the idea of capabilities of numerous advanced variables, symplectic geometry, h-principles, and Morse idea that input the proofs of the most effects. the most result of the ebook are unique result of the authors, and several other of those effects look right here for the 1st time. The booklet might be precious for all scholars and mathematicians drawn to geometric facets of complicated research, symplectic and make contact with topology, and the interconnections among those matters.
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Discontinuous Groups of Isometries in the Hyperbolic Plane
This e-book via Jakob Nielsen (1890-1959) and Werner Fenchel (1905-1988) has had
a lengthy and intricate historical past. In 1938-39, Nielsen gave a sequence of lectures on
discontinuous teams of motions within the non-euclidean airplane, and this led him - in the course of
World battle II - to jot down the 1st chapters of the ebook (in German). whilst Fenchel,
who needed to get away from Denmark to Sweden as a result German profession,
returned in 1945, Nielsen initiated a collaboration with him on what grew to become identified
as the Fenchel-Nielsen manuscript. at the moment they have been either on the Technical
University in Copenhagen. the 1st draft of the Fenchel-Nielsen manuscript (now
in English) was once complete in 1948 and it was once deliberate to be released within the Princeton
Mathematical sequence. notwithstanding, a result of fast improvement of the topic, they felt
that colossal alterations needed to be made earlier than booklet.
When Nielsen moved to Copenhagen college in 1951 (where he stayed until eventually
1955), he was once a lot concerned with the foreign association UNESCO, and the
further writing of the manuscript was once left to Fenchel. The records of Fenchel now
deposited and catalogued on the division of arithmetic at Copenhagen Univer-
sity include unique manuscripts: a partial manuscript (manuscript zero) in Ger-
man containing Chapters I-II (
I -15), and a whole manuscript (manuscript I) in
English containing Chapters I-V (
1-27). The information additionally comprise a part of a corre-
spondence (first in German yet later in Danish) among Nielsen and Fenchel, the place
Nielsen makes precise reviews to Fenchel's writings of Chapters III-V. Fenchel,
who succeeded N. E. Nf/Jrlund at Copenhagen college in 1956 (and stayed there
until 1974), used to be a great deal concerned with a radical revision of the curriculum in al-
gebra and geometry, and focused his study within the idea of convexity, heading
the overseas Colloquium on Convexity in Copenhagen 1965. for nearly two decades
he additionally positioned a lot attempt into his task as editor of the newly begun magazine Mathematica
Scandinavica. a lot to his dissatisfaction, this task left him little time to complete the
Fenchel-Nielsen undertaking the best way he desired to.
After his retirement from the college, Fenchel - assisted via Christian Sieben-
eicher from Bielefeld and Mrs. Obershelp who typed the manuscript - came across time to
finish the e-book undemanding Geometry in Hyperbolic area, which was once released via
Walter de Gruyter in 1989 presently after his loss of life. at the same time, and with a similar
collaborators, he supervised a typewritten model of the manuscript (manuscript 2) on
discontinuous teams, elimination the various vague issues that have been within the unique
manuscript. Fenchel advised me that he pondered elimination elements of the introductory
Chapter I within the manuscript, because this may be coated via the e-book pointed out above;
but to make the Fenchel-Nielsen publication self-contained he eventually selected to not do
so. He did choose to pass over
27, entitled Thefundamental team.
As editor, i began in 1990, with the consent of the felony heirs of Fenchel and
Nielsen, to supply a TEX-version from the newly typewritten model (manuscript 2).
I am thankful to Dita Andersen and Lise Fuldby-Olsen in my division for hav-
ing performed an excellent task of typing this manuscript in AMS- TEX. i've got additionally had
much support from my colleague J0rn B0rling Olsson (himself a scholar of Kate Fenchel
at Aarhus college) with the evidence examining of the TEX-manuscript (manuscript three)
against manuscript 2 in addition to with a common dialogue of the variation to the fashion
of TEX. In such a lot respects we made up our minds to persist with Fenchel's intentions. even though, turning
the typewritten version of the manuscript into TEX helped us to make sure that the notation,
and the spelling of definite key-words, will be uniform through the ebook. additionally,
we have indicated the start and finish of an explanation within the traditional variety of TEX.
With this TEX -manuscript I approached Walter de Gruyter in Berlin in 1992, and
to my nice reduction and pride they agreed to submit the manuscript of their sequence
Studies in arithmetic. i'm so much thankful for this confident and speedy response. One
particular challenge with the ebook became out to be the copy of the numerous
figures that are an essential component of the presentation. Christian Siebeneicher had at
first agreed to bring those in ultimate digital shape, yet via 1997 it grew to become transparent that he
would now not be ready to locate the time to take action. besides the fact that, the writer provided an answer
whereby I should still carry exact drawings of the figures (Fenchel didn't depart such
for Chapters IV and V), after which they'd arrange the creation of the figures in
electronic shape. i'm very thankful to Marcin Adamski, Warsaw, Poland, for his advantageous
collaboration in regards to the real creation of the figures.
My colleague Bent Fuglede, who has personaHy identified either authors, has kindly
written a brief biography of the 2 of them and their mathematical achievements,
and which additionally locations the Fenchel-Nielsen manuscript in its right standpoint. In
this connection i need to thank The Royal Danish Academy of Sciences and
Letters for permitting us to incorporate during this booklet reproductions of photos of the 2
authors that are within the ownership of the Academy.
Since the manuscript makes use of a few detailed symbols, a listing of notation with brief
explanations and connection with the particular definition within the booklet has been incorporated. additionally,
a finished index has been further. In either situations, all references are to sections,
not pages.
We thought of including an entire checklist of references, yet determined opposed to it because of
the overwhelming variety of examine papers during this quarter. as a substitute, a far shorter
list of monographs and different complete debts correct to the topic has been
collected.
My ultimate and such a lot honest thank you visit Dr. Manfred Karbe from Walter de Gruyter
for his commitment and perseverance in bringing this e-book into life.
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Extra info for From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds
Sample text
This condition ensures that the boundary of the open cone Kp := {z ∈ Uj ∩ W | fp (z) > 0} in W is given by {z ∈ Uj ∩ W | fp (z) = 0} for all p ∈ Wj ∩ L. To see this, suppose by contradiction that there exists a point z = (u, v) ∈ K p ∩ ∂Uj . Then 2|v|2 − |u − p|2 = fp (z) ≥ 0 and the choice of W yield the contradiction dist(Wj , ∂Uj )2 ≤ |u − p|2 + |v|2 ≤ 3|v|2 < dist(Wj , ∂Uj )2 . Moreover, we can choose W so small that fp is J-convex on Kp for all p ∈ Wj ∩ W . Let φp := g ◦ fp , where g(t) = e−1/t for t > 0 and 0 for t ≤ 0.
A) Adding up ddC (βψ) = β ddC ψ + dβ ∧ dC ψ + dψ ∧ dC β + ψ ddC β and the corresponding equation for (1 − β)φ at any point x ∈ V , we find −ddC ϑ = −(1 − β) ddC φ − βddC ψ + dβ ∧ dC (φ − ψ) + d(φ − ψ) ∧ dC β + (φ − ψ)ddC β ≥ min (mφ , mψ ) − 2|dβ| |d(φ − ψ)| − |φ − ψ| |ddC β| > 0. (b) At the point x the terms φ − ψ and dφ − dψ vanish, so the computation in (a) shows −ddC ϑ = −(1−β) ddC φ−βddC ψ. Hence at the point x we have dϑ = dφ = dψ and the associated metrics satisfy gϑ = (1 − β)gφ + βgψ .
1) holds} is called the modulus of subharmonicity of the function φ. 1) for functions δ supported near z. 1), then choosing a sequence of functions δn converging to the Dirac measure of a point z ∈ U and integrating by parts shows ∆φ(z) ≥ m(z), so for a C 2 -function the two definitions agree and mφ = ∆φ. 1By “subharmonic” we will always mean “strictly subharmonic”. Non-strict subharmonicity will be referred to as “weak subharmonicity”. The same applies to plurisubharmonicity discussed below. 33 34 3.