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Theses

Aspects twistoriels des applications semi-conformes

Abstract : A semi-conformal mapping generalises the notion of conformal mapping in the plane to higher dimensions. In the first part of the thesis a rigourous examination of the theory of semi-conformal mappings is undertaken. It is shown how they arise as extremals of a functional, how they satisfy a conservation law about a singularity and how certain families of biharmonic semi-conformal mappings are classied in terms of holomorphic data. All of this in new and already makes a signicant contribution to the subject. This prepares the ground for the subsequent chapters. The key results that follow are (i) a generalization of a result of J. C.Wood to space-time, specically, any null solution of the wave-equation has exactly two null directions in its kernel, at least one of which is geodesic and shear-free; (ii) the construction of a coupled evolution of a Riemannian metric and a eld in 3 dimensions in such a way the the eld remains semi-conformal; this gives an original unconventional way in which to describe massless elds in a more general curved space-time, furthermore it prepares the ground for the fourth chapter where an analogeous evolution on a graph is presented; (iii) the description of the harmonic map heat ow on a nite graph, proving necessary and sufficient conditions for both long term existence and convergence; (iv) the development of twistor theory on a finite graph, in particular, We introduces the twistor correspondence in terms of the line-graph of a given graph.
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Submitted on : Wednesday, March 3, 2010 - 3:41:59 PM
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Mohammad Wehbe. Aspects twistoriels des applications semi-conformes. Mathématiques [math]. Université de Bretagne occidentale - Brest, 2009. Français. ⟨tel-00461149⟩

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