Hebrew University Topology and Geometry Seminar

April 6, 2016
Special place: Levi building, Room 06

Sari Ghanem

Université Joseph Fourier - Grenoble I

The decay of SU(2) Yang-Mills fields on the Schwarzschild black hole with spherically symmetric small energy initial data

Abstract: The talk will be aimed at introducing the subject of mathematical General Relativity to a broad audience, and at presenting some of the main progress and famous open problems in the field. In this context of mathematical challenges in General Relativity, after having explained the basic background and motivation, I will present the Yang-Mills equations on arbitrary fixed curved space-times, valued in the Lie algebra associated to any arbitrary Lie group. Thereafter, I will expose recent results with Dietrich Häfner concerning the Yang-Mills fields valued in the Lie algebra su(2) associated to the Lie group SU(2), propagating on the Schwarzschild black hole. We assume that the initial data are spherically symmetric satisfying a certain Ansatz, and have small energy, which eliminates the stationary solutions which do not decay. We then prove uniform decay estimates in the entire exterior region of the black hole, including the event horizon, for gauge invariant norms on the Yang-Mills curvature generated from such initial data, including the $ L^\infty $ norm of the so-called middle components. This is done by proving a Morawetz type estimate that is stronger than the one assumed in previous work, without passing through the scalar wave equation on the Yang-Mills curvature, using the Yang-Mills equations directly.