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# Geometric Analysis and Partial Differential Equations Seminar

## September 25, 2014 @ 10:30 am - 11:30 pm

##### Speaker

Dr Stephen McCormick (University of New England)

##### Title

The Phase Space for the Einstein-Yang-Mills Equations and the First Law of Black Hole Thermodynamics

##### Abstract

The Einstein-Yang-Mills constraint equations form a system of “almost elliptic” nonlinear PDE, describing the gravitational field coupled to gauge fields. In this talk we provide a Hilbert manifold structure for the space of solutions to this system on an asymptotically flat manifold; this ensures the validity of first-order perturbation theory for the system. Using this Hilbert manifold structure, we demonstrate that the first law of black hole mechanics gives a condition for a solution to be stationary.