geometric evolution equations; local monotonicity; Yang-Mills flow; harmonic map heat flow; mean curvature flow
The aim of this thesis is to establish local monotonicity formulæ for solutions to Dirichlet-type flows, such as the harmonic map and Yang-Mills heat flows, and the mean curvature flow. In particular, for the former, we allow as domain an evolving Riemannian manifold and for the latter, we allow as target an evolving Riemannian manifold. The approach taken consists in first deriving divergence identities involving an appropriate evolving quantity, then integrating over superlevel sets (heat balls) of suitable kernels. A theory of heat balls analogous to that of Ecker, Knopf, Ni and Topping is developed in order to accomplish this. The main result is then that, provided certain integrals are finite, local monotonicity formulæ hold in this general setting, thus generalizing results for the mean curvature and harmonic map heat flows and establishing a new local monotonicity formula for solutions to the Yang-Mills flow.
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