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Conformal harmonic forms, Branson–Gover operators and Dirichlet problem at infinity

Erwann Aubry, Colin Guillarmou (2011)

Journal of the European Mathematical Society

For odd-dimensional Poincaré–Einstein manifolds ( X n + 1 , g ) , we study the set of harmonic k -forms (for k < n / 2 ) which are C m (with m ) on the conformal compactification X ¯ of X . This set is infinite-dimensional for small m but it becomes finite-dimensional if m is large enough, and in one-to-one correspondence with the direct sum of the relative cohomology H k ( X ¯ , X ¯ ) and the kernel of the Branson–Gover [3] differential operators ( L k , G k ) on the conformal infinity ( X ¯ , [ h 0 ] ) . We also relate the set of C n - 2 k + 1 ( Λ k ( X ¯ ) ) forms in the kernel of d + δ g to the conformal...

Conformally bending three-manifolds with boundary

Matthew Gursky, Jeffrey Streets, Micah Warren (2010)

Annales de l’institut Fourier

Given a three-dimensional manifold with boundary, the Cartan-Hadamard theorem implies that there are obstructions to filling the interior of the manifold with a complete metric of negative curvature. In this paper, we show that any three-dimensional manifold with boundary can be filled conformally with a complete metric satisfying a pinching condition: given any small constant, the ratio of the largest sectional curvature to (the absolute value of) the scalar curvature is less than this constant....

Conjugate-cut loci and injectivity domains on two-spheres of revolution

Bernard Bonnard, Jean-Baptiste Caillau, Gabriel Janin (2013)

ESAIM: Control, Optimisation and Calculus of Variations

In a recent article [B. Bonnard, J.-B. Caillau, R. Sinclair and M. Tanaka, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009) 1081–1098], we relate the computation of the conjugate and cut loci of a family of metrics on two-spheres of revolution whose polar form is g = dϕ2 + m(ϕ)dθ2 to the period mapping of the ϕ-variable. One purpose of this article is to use this relation to evaluate the cut and conjugate loci for a family of metrics arising as a deformation of the round sphere and to determine...

Conley index in Hilbert spaces and a problem of Angenent and van der Vorst

Marek Izydorek, Krzysztof P. Rybakowski (2002)

Fundamenta Mathematicae

In a recent paper [9] we presented a Galerkin-type Conley index theory for certain classes of infinite-dimensional ODEs without the uniqueness property of the Cauchy problem. In this paper we show how to apply this theory to strongly indefinite elliptic systems. More specifically, we study the elliptic system - Δ u = v H ( u , v , x ) in Ω, - Δ v = u H ( u , v , x ) in Ω, u = 0, v = 0 in ∂Ω, (A1) on a smooth bounded domain Ω in N for "-"-type Hamiltonians H of class C² satisfying subcritical growth assumptions on their first order derivatives....

Connecting topological Hopf singularities

Robert Hardt, Tristan Rivière (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Smooth maps between riemannian manifolds are often not strongly dense in Sobolev classes of finite energy maps, and an energy drop in a limiting sequence of smooth maps often is accompanied by the production (or bubbling) of an associated rectifiable current. For finite 2-energy maps from the 3 ball to the 2 sphere, this phenomenon has been well-studied in works of Bethuel-Brezis-Coron and Giaquinta-Modica-Soucek where a finite mass 1 dimensional rectifiable current occurs whose boundary is the...

Connections for non-holonomic 3-webs

Vanžurová, Alena (1997)

Proceedings of the 16th Winter School "Geometry and Physics"

A non-holonomic 3-web is defined by two operators P and B such that P is a projector, B is involutory, and they are connected via the relation P B + B P = B . The so-called parallelizing connection with respect to which the 3-web distributions are parallel is defined. Some simple properties of such connections are found.

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