Hyperbolic Multi-Monopoles With Arbitrary Mass

On a complete manifold, such as Euclidean 3-space or hyperbolic 3-space, the limit at infinity of the norm of the Higgs field is called the mass of the monopole. We show the existence, on hypebolic 3-space, of monopoles with given magnetic charge and arbitrary mass. Previously, aside from charge one monopoles, existence was known only for monopoles with integral mass (since these arise from U(1) invariant instantons on Euclidean 4-space). The method of proof is based on Taubes' gluing procedure, using well-separated, explicit, charge one monopoles. The analysis is carried out in a weighted Sobolev space and necessitates eliminating the possibility of point spectra.Comment: 20 page

A compactness theorem for complete Ricci shrinkers

We prove precompactness in an orbifold Cheeger-Gromov sense of complete gradient Ricci shrinkers with a lower bound on their entropy and a local integral Riemann bound. We do not need any pointwise curvature assumptions, volume or diameter bounds. In dimension four, under a technical assumption, we can replace the local integral Riemann bound by an upper bound for the Euler characteristic. The proof relies on a Gauss-Bonnet with cutoff argument.Comment: 28 pages, final version, to appear in GAF