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Theses

Équation de Monge-Ampère complexe, métriques kählériennes de type Poincaré et instantons gravitationnels ALF

Abstract : The topic of this thesis is the resolution of Monge-Ampère equations and its application on certain types of non-compact manifolds. This dissertation describes more precisely two distinct situations in which we solve Monge-Ampère equations, and draw conclusions of these resolutions. We work in a first part on the complement of a divisor with normal crossings in a compact Kähler manifold. We fix on the complement of the divisor a class of Kähler metrics with cusp singularities along the divisor. In order to construct geodesics joining metrics of this class, we solve a homogeneous Monge-Ampère equation on the product of our Zariski open set with some Riemann surface with boundary. This construction is then applied to a uniqueness results for constant scalar curvature metrics in the considered class; for this, we also solve a Monge-Ampère equation with right-hand-side member on the complement of the divisor. We finally prove topological obstructions to the existence of constant scalar curvature metrics among the classes of singular metrics we are interested in. The second part of the dissertation is devoted to an analytic construction of ALF gravitational instantons, that is complete hyperkähler 4-manifolds, with cubic growth of the volume. We give the construction of some dihedral instantons ; more specifically, we consider resolutions of kleinian singularities. The treatment of a Monge-Ampère equation, given for quite general ALF manifolds, allows us to correct on our examples a simple prototype to get the sought hyperkähler metric.
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https://tel.archives-ouvertes.fr/tel-00750891
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Submitted on : Tuesday, November 13, 2012 - 9:45:31 AM
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Hugues Auvray. Équation de Monge-Ampère complexe, métriques kählériennes de type Poincaré et instantons gravitationnels ALF. Géométrie différentielle [math.DG]. Université Pierre et Marie Curie - Paris VI, 2012. Français. ⟨tel-00750891⟩

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