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Register a new userLimits of Yang-Mills {alpha}-connections
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Authors
Casey Lynn Kelleher
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In the spirit of recent work of Lamm, Malchiodi and Micallef in the setting of harmonic maps, we identify Yang-Mills connections obtained by approximations with respect to the Yang-Mills {alpha}-energy. More specifically, we show that for the SU(2) Hopf fibration over the four sphere, for sufficiently small {alpha} values the SO(4) invariant ADHM instanton is the unique {alpha}-critical point which has Yang-Mills {alpha}-energy lower than a specific threshold.
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On a Riemannian manifold of dimension $n$ we extend the known analytic results on Yang-Mills connections to the class of connections called $Omega$-Yang-Mills connections, where $Omega$ is a smooth, not necessarily closed, $(n-4)$-form. Special cases include $Omega$-anti-self-dual connections and Hermitian-Yang-Mills connections over general complex manifolds. By a key observation, a weak compactness result is obtained for moduli space of smooth $Omega$-Yang-Mills connections with uniformly $L^2$ bounded curvature, and it can be improved in the case of Hermitian-Yang-Mills connections over general complex manifolds. A removable singularity theorem for singular $Omega$-Yang-Mills connections on a trivial bundle with small energy concentration is also proven. As an application, it is shown how to compactify the moduli space of smooth Hermitian-Yang-Mills connections on unitary bundles over a class of balanced manifolds of Hodge-Riemann type. This class includes the metrics coming from multipolarizations, and in particular, the Kaehler metrics. In the case of multipolarizations on a projective algebraic manifold, the compactification of smooth irreducible Hermitian-Yang-Mills connections with fixed determinant modulo gauge transformations inherits a complex structure from algebro-geometric considerations.
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