Regularity and vanishing theorems for Yang–Mills–Higgs pairs

Abstract

We obtain a vanishing theorem for Yang–Mills–Higgs pairs on Euclidean and hyperbolic spaces in dimensions greater than 4, as well as a regularity theorem more generally on Riemannian manifolds, and show how the latter may be used to establish a partial compactness theorem. Along the way, we investigate the monotonicity and scaling properties of such pairs, obtaining a generalisation of the well-known monotonicity formula for Yang-Mills connections.