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A nonlinear Poisson transform for Einstein metrics on product spaces

Olivier Biquard, Rafe Mazzeo (2011)

Journal of the European Mathematical Society

We consider the Einstein deformations of the reducible rank two symmetric spaces of noncompact type. If M is the product of any two real, complex, quaternionic or octonionic hyperbolic spaces, we prove that the family of nearby Einstein metrics is parametrized by certain new geometric structures on the Furstenberg boundary of M .

An index inequality for embedded pseudoholomorphic curves in symplectizations

Michael Hutchings (2002)

Journal of the European Mathematical Society

Let Σ be a surface with a symplectic form, let φ be a symplectomorphism of Σ , and let Y be the mapping torus of φ . We show that the dimensions of moduli spaces of embedded pseudoholomorphic curves in × 𝕐 , with cylindrical ends asymptotic to periodic orbits of φ or multiple covers thereof, are bounded from above by an additive relative index. We deduce some compactness results for these moduli spaces. This paper establishes some of the foundations for a program with Michael Thaddeus, to understand...

Asymptotic behaviour and the moduli space of doubly-periodic instantons

Olivier Biquard, Marcos Jardim (2001)

Journal of the European Mathematical Society

We study doubly-periodic instantons, i.e. instantons on the product of a 1-dimensional complex torus T with a complex line , with quadratic curvature decay. We determine the asymptotic behaviour of these instantons, constructing new asymptotic invariants. We show that the underlying holomorphic bundle extends to T × 1 . The converse statement is also true, namely a holomorphic bundle on T × 1 which is flat on the torus at infinity, and satisfies a stability condition, comes from a doubly-periodic instanton....

Classification analytique de structures de Poisson

Philipp Lohrmann (2009)

Bulletin de la Société Mathématique de France

Notre étude porte sur une catégorie de structures de Poisson singulières holomorphes au voisinage de 0 n et admettant une forme normale formelle polynomiale i.e. un nombre fini d’invariants formels. Les séries normalisantes sont divergentes en général. On montre l’existence de transformations normalisantes holomorphes sur des domaines sectoriels de la forme a < arg x R < b , où x R est un monôme associé au problème. Il suit une classification analytique.

Generating varieties for the triple loop space of classical Lie groups

Yasuhiko Kamiyama (2003)

Fundamenta Mathematicae

For G = SU(n), Sp(n) or Spin(n), let C G ( S U ( 2 ) ) be the centralizer of a certain SU(2) in G. We have a natural map J : G / C G ( S U ( 2 ) ) Ω ³ G . For a generator α of H ( G / C G ( S U ( 2 ) ) ; / 2 ) , we describe J⁎(α). In particular, it is proved that J : H ( G / C G ( S U ( 2 ) ) ; / 2 ) H ( Ω ³ G ; / 2 ) is injective.

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...

Infinitesimal conjugacies and Weil-Petersson metric

Albert Fathi, L. Flaminio (1993)

Annales de l'institut Fourier

We study deformations of compact Riemannian manifolds of negative curvature. We give an equation for the infinitesimal conjugacy between geodesic flows. This in turn allows us to compute derivatives of intersection of metrics. As a consequence we obtain a proof of a theorem of Wolpert.

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