MongeAmpere equation and CalabiYau manifolds:
talks and abstracts
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Zbigniew Blocki (Krakow)
The CalabiYau Theorem
We will sketch the proof of the CalabiYau theorem
which asserts that on a compact Kahler manifold M the map sending
a Kahler metric from a given Kahler class to its Ricci curvature,
which is an element of the first Chern class of M, is bijective.
It is equivalent to proving that every positive volume form on M
with a fixed total volume is a Kahler volume of a unique Kahler
metric (in a given Kahler class), and this boils down to solving
the complex MongeAmpere equation. Building up on the work of
Calabi, Nirenberg and Aubin, ShingTung Yau finished the proof
of this result in 1976, for which he was awarded the Fields Medal
in 1982. One of the amazing consequences is existence of
Ricciflat metrics on manifolds with vanishing first Chern class.
Except for toruses, these metrics can never be written down
explicitly!
Literature:

The complex MongeAmpere equation in Kahler geometry,
course given at CIME Summer School in Pluripotential
Theory, Cetraro, Italy, July 2011, eds. F. Bracci,
J. E. Fornaess, Lecture Notes in Mathematics 2075,
pp. 95142, Springer, 2013

The CalabiYau theorem,
course given at the Winter School in Complex Analysis,
Toulouse, January 2005, appeared in "Complex MongeAmpere
equations and geodesics in the space of Kahler metrics",
ed. V. Guedj, Lecture Notes in Mathematics 2038,
pp. 201227, Springer, 2012.
Zbigniew Blocki (Krakow)
The space of Kahler metrics
Mabuchi, Semmes and Donaldson have independently
introduced a natural Riemannian structure on (infitely dimensional)
space of Kahler metrics on a compact manifold fo dimension n. It
turns out that geodesics in this space are solutions of the
homogeneous complex MongeAmpere equation in the space of
dimension n+1. X.X.Chen proved existence of weak almost C^{1,1}
geodesics and Lempert and Vivas showed that they need not be smooth
in general. Nevertheless, these geodesics play crucial role
in the proof of uniqueness of constant scalar curvature metrics
modulo holomorphic automorphism in a given Kahler class.
Slawomir Dinew (Krakow)
Local regularity of the complex MongeAmpere equation
We shall review recent results on the local regularity of the complex MongeAmpere
equation. In particular we shall be interested in the
problem of optimal conditions
implying that local solutions of the equation with "good" data are smooth. Several illustrating
examples will be shown. We shall also compare the existent results with the analogous
real MongeAmpere theory due to Caffarelli.
Jakob
Hultgren (Chalmers)
Coupled KahlerEinstein Metrics
A central theme in complex geometry has been to study various types of canonical metrics,
for example KahlerEinstein metrics and cscK metrics. In this talk we will introduce the
notion of coupled KahlerEinstein (cKE) metrics which are
ktuples of Kahler metrics that satisfy certain coupled
KahlerEinstein equations. We will discuss existence and
uniqueness properties and elaborate on related algebraic
stability conditions. (Joint work with David Witt Nystrom)
Mattias Jonsson (Ann Arbor)
On degenerations of volume forms and CalabiYau varieties
Various considerations in mirror symmetry and moduli
problems leads one to study degenerating oneparameter
families of complex manifolds. For example, a conjecture
by KontsevichSoibelman describes the GromovHausdorff
limits of certain degenerations of CalabiYau manifolds.
I will give an introduction to these circle of ideas, and
then present joint work with Sebastien Boucksom, where we
study the asymptotic behavior of volume forms (i.e. smooth
measures) on a degenerating family of compact complex
manifolds. Under rather general conditions, we prove that
the volume forms converge in a natural sense to a
Lebesguetype measure on a certain simplicial complex.
Alexander Kolesnikov (HSE)
Hessian metrics related to mass transportation problem. Applications
to geometry and analysis
The MongeKantorovich šoptimal mass transporation problem
has numerous applications in analysis,
probability, and PDE's. We will talk about its interplay
with the differential geometry, šfocusing
the metric properties of the related Hessian space with
convex potential solving the corresponding real
MongeAmpere equation. We discuss recent extensions of
some classical results on the KaehlerEinstein
equation and explain thešmotivation coming from convex
geometry and probability.š
Slawomir Kolodziej (Krakow)
Pluripotential methods of solving the complex
MongeAmpere equation
The aim of the minicourse is to show how the pluripotential methods
can be applied to prove existence and stability of weak solutions of the
MongeAmpere equation on compact Kahler or just Hermitian manifolds.
First the basic notions and results of pluripotential theory (after Bedford
and Taylor) are discussed. Using them we will sketch the proof of a priori
estimates and the existence of solutions of the complex MongeAmpere
equation on Kahler manifolds when the right hand side is in Lp(X, n),
p > 1. Then we adapt the methods to the Hermitian
case.
For background material see the first chapter of:
S. Kolodziej, The complex MongeAmpere equation and pluripotential
theory, Memoirs Amer. Math. Soc. 178 (2005), pp. 64.
Magnus Önnheim (Göteborg)
MongeAmpere equations on compact Hessian manifold
In the study of complex MongeAmpere equations,
variational formulations have managed to produce several
existence and uniqueness results. In the setting of
Euclidean space, a variational formulation of the real
inhomogeneous MongeAmpere equation is given by the
Kantorovich functional from optimal transport (analogous
to the Ding functional in complex geometry).
In this talk, inspired by ideas of Shima and Gross, we
show that the Legendre transform can be given a natural
geometric meaning in the setting of a Hessian manifold. We
proceed to show that using the Legendre transform to
define a Kantorovich type functional we obtain a
variational formulation for inhomogeneous MongeAmpere
equations on M. One feature of this approach is that it
provides a way to handle very singular equations. If time
permits we briefly discuss some interesting phenomena
relating to this. (Joint work with Jakob Hultgren)
Vladimir Roubtsov (Angers)
MongeAmpere equations, geometric structures
and meteorological hydrodynamics
We describe how the geometric approach to MongeAmpere
operators and equations proposed by V.Lychagin
might be used in (quasi)geostrophic hydrodynamical
models. Some naturally arised geometric structures
(Kahler, Hyperkahler/symplectic, Generalized Complex...)
in this models are discussed.
Jian Song (Rutgers)
The KahlerRicci flow through singularities
We study the formation of finite time singularities for
the KahlerRicci flow in relation to birational surgeries
such as divisorial contractions and flips. We show that
the KahlerRicci flow can be uniquely continued through
such birational surgeries. In particular, we construct a
global family of flips by the KahlerRicci flow in
GromovHausdorff topology.
Yuguang Zhang (Beijing)
Balanced embedding of degenerating Abelian varieties
In this talk, I show that, after a base change, a certain degeneration
of Abelian varieties can be simultaneously balanced
embedded into a projective
space by the theta functions. Then I study the
relationship between the balanced central fiber and
the GromovHausdorff limit of flat Kahler metrics on the nearby fibers.
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