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Aproximation of Z2-cocycles and shift dynamical systems.

I. Filipowicz, J. Kwiatkowski, M. Lemanczyk (1988)

Publicacions Matemàtiques

Let Gbar = G{nt, nt | nt+1, t ≥ 0} be a subgroup of all roots of unity generated by exp(2πi/nt}, t ≥ 0, and let τ: (X, β, μ) O be an ergodic transformation with pure point spectrum Gbar. Given a cocycle φ, φ: X → Z2, admitting an approximation with speed 0(1/n1+ε, ε>0) there exists a Morse cocycle φ such that the corresponding transformations τφ and τψ are relatively isomorphic. An effective way of a construction of the Morse cocycle φ is given. There is a cocycle φ oddly approximated with...

Concentration phenomena of two-vortex solutions in a Chern-Simons model

Chiun-Chuan Chen, Chang-Shou Lin, Guofang Wang (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

By considering an abelian Chern-Simons model, we are led to study the existence of solutions of the Liouville equation with singularities on a flat torus. A non-existence and degree counting for solutions are obtained. The former result has an application in the Chern-Simons model.

Extrema de valeurs propres dans une classe conforme

Pierre Jammes (2005/2006)

Séminaire de théorie spectrale et géométrie

On s’intéresse au problème de savoir quelle est la rigidité apportée au spectre d’une variété riemannienne compacte par le fait de fixer son volume et se classe conforme, et en particulier de déterminer si on peut faire tendre les valeurs propres vers 0 ou l’infini sous cette contrainte. On considère successivement les cas du laplacien usuel agissant sur les fonctions, l’opérateur de Dirac, le laplacien conforme et le laplacien de Hodge-de Rham.

Heat flows for extremal Kähler metrics

Santiago R. Simanca (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let ( M , J , Ω ) be a closed polarized complex manifold of Kähler type. Let G be the maximal compact subgroup of the automorphism group of ( M , J ) . On the space of Kähler metrics that are invariant under G and represent the cohomology class Ω , we define a flow equation whose critical points are the extremal metrics,i.e.those that minimize the square of the L 2 -norm of the scalar curvature. We prove that the dynamical system in this space of metrics defined by the said flow does not have periodic orbits, and that its...

Infinite geodesic rays in the space of Kähler potentials

Claudio Arezzo, Gang Tian (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper we prove the existence of solutions of a degenerate complex Monge-Ampére equation on a complex manifold. Applying our existence result to a special degeneration of complex structure, we show how to associate to a change of complex structure an infinite length geodetic ray in the space of potentials. We also prove an existence result for the initial value problem for geodesics. We end this paper with a discussion of a list of open problems indicating how to relate our reults to the...

Nontrivial examples of coupled equations for Kähler metrics and Yang-Mills connections

Julien Keller, Christina Tønnesen-Friedman (2012)

Open Mathematics

We provide nontrivial examples of solutions to the system of coupled equations introduced by M. García-Fernández for the uniformization problem of a triple (M; L; E), where E is a holomorphic vector bundle over a polarized complex manifold (M, L), generalizing the notions of both constant scalar curvature Kähler metric and Hermitian-Einstein metric.

On the Moser-Onofri and Prékopa-Leindler inequalities.

Alessandro Ghigi (2005)

Collectanea Mathematica

Using elementary convexity arguments involving the Legendre transformation and the Prékopa-Leindler inequality, we prove the sharp Moser-Onofri inequality, which says that1/16π ∫|∇φ|2 + 1/4π ∫ φ - log (1/4π ∫ eφ) ≥ 0for any funcion φ ∈ C∞(S2).

On the Paneitz energy on standard three sphere

Paul Yang, Meijun Zhu (2004)

ESAIM: Control, Optimisation and Calculus of Variations

We prove that the Paneitz energy on the standard three-sphere S 3 is bounded from below and extremal metrics must be conformally equivalent to the standard metric.

On the Paneitz energy on standard three sphere

Paul Yang, Meijun Zhu (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We prove that the Paneitz energy on the standard three-sphere S3 is bounded from below and extremal metrics must be conformally equivalent to the standard metric.

Some critical almost Kähler structures

Takashi Oguro, Kouei Sekigawa (2008)

Colloquium Mathematicae

We consider the set of all almost Kähler structures (g,J) on a 2n-dimensional compact orientable manifold M and study a critical point of the functional ℱ λ , μ ( J , g ) = ∫ M ( λ τ + μ τ * ) d M g with respect to the scalar curvature τ and the *-scalar curvature τ*. We show that an almost Kähler structure (J,g) is a critical point of ℱ - 1 , 1 if and only if (J,g) is a Kähler structure on M.

The Calabi functional on a ruled surface

Gábor Székelyhidi (2009)

Annales scientifiques de l'École Normale Supérieure

We study the Calabi functional on a ruled surface over a genus two curve. For polarizations which do not admit an extremal metric we describe the behavior of a minimizing sequence splitting the manifold into pieces. We also show that the Calabi flow starting from a metric with suitable symmetry gives such a minimizing sequence.

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