1,016 research outputs found
K\"ahler immersions of K\"ahler manifolds into complex space forms
The study of K\"ahler immersions of a given real analytic K\"ahler manifold
into a finite or infinite dimensional complex space form originates from the
pioneering work of Eugenio Calabi [10]. With a stroke of genius Calabi defines
a powerful tool, a special (local) potential called diastasis function, which
allows him to obtain necessary and sufficient conditions for a neighbourhood of
a point to be locally K\"ahler immersed into a finite or infinite dimensional
complex space form. As application of its criterion, he also provides a
classification of (finite dimensional) complex space forms admitting a K\"ahler
immersion into another. Although, a complete classification of K\"ahler
manifolds admitting a K\"ahler immersion into complex space forms is not known,
not even when the K\"ahler manifolds involved are of great interest, e.g. when
they are K\"ahler-Einstein or homogeneous spaces. In fact, the diastasis
function is not always explicitely given and Calabi's criterion, although
theoretically impeccable, most of the time is of difficult application.
Nevertheless, throughout the last 60 years many mathematicians have worked on
the subject and many interesting results have been obtained.
The aim of this book is to describe Calabi's original work, to provide a
detailed account of what is known today on the subject and to point out some
open problems.Comment: 116 page
Some remarks on homogeneous K\"ahler manifolds
In this paper we provide a positive answer to a conjecture due to A. J. Di
Scala, A. Loi, H. Hishi (see [3, Conjecture 1]) claiming that a
simply-connected homogeneous K\"ahler manifold M endowed with an integral
K\"ahler form , admits a holomorphic isometric immersion in the
complex projective space, for a suitable . This result has two
corollaries which extend to homogeneous K\"ahler manifolds the results obtained
by the authors in [8] and in [12] for homogeneous bounded domains.Comment: 8 page
Compact Stein surfaces with boundary as branched covers of \B^4
We prove that Stein surfaces with boundary coincide up to orientation
preserving diffeomorphisms with simple branched coverings of \B^4 whose
branch set is a positive braided surface. As a consequence, we have that a
smooth oriented 3-manifold is Stein fillable iff it has a positive open-book
decomposition.Comment: 25 pages, 20 postscript figures. LaTeX file. Uses: geom.sty epsf.st
Evolution paths on the equilibrium manifold
In a pure exchange smooth economy with fixed total resources, we de- fine the length between two regular equilibria belonging to the equilibrium manifold as the number of intersection points of the evolution path connecting them with the set of critical equilibria. We show that there exists a minimal path according to this definition of length.Equilibrium manifold; regular economies; critical equilibria; catastrophes; Jordan-Brouwer separation theorem
A note on the structural stability of the equilibrium manifold
In a smooth pure exchange economy with fixed total resources we investigate whether the smooth selection property holds when endowments are redistributed across consumers through a continuous (non local) redistribution policy. We show that if the policy is regular then there exists a unique continuous path of equilibrium prices which support it. If singular economies are involved in the redistribution, then an analogous result can be obtained if the singular policy is the projection of a path transversal to the set of critical equilibria.Equilibrium manifold, regular equilibria, catastrophes, structural stability, smooth selection of prices
- …