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

Nonlinear Hodge maps

We consider maps between Riemannian manifolds in which the map is a stationary point of the nonlinear Hodge energy. The variational equations of this functional form a quasilinear, nondiagonal, nonuniformly elliptic system which models certain kinds of compressible flow. Conditions are found under which singular sets of prescribed dimension cannot occur. Various degrees of smoothness are proven for the sonic limit, high-dimensional flow, and flow having nonzero vorticity. The gradient flow of solutions is estimated. Implications for other quasilinear field theories are suggested.Comment: Slightly modified and updated version; tcilatex, 32 page

Duality methods for a class of quasilinear systems

Duality methods are used to generate explicit solutions to nonlinear Hodge systems, demonstrate the well-posedness of boundary value problems, and reveal, via the Hodge-B\"acklund transformation, underlying symmetries among superficially different forms of the equations.Comment: 14 page

Anti-self-dual instantons with Lagrangian boundary conditions II: Bubbling

We study bubbling phenomena of anti-self-dual instantons on \H^2\times\S, where $\S$ is a closed Riemann surface. The restriction of the instanton to each boundary slice $\{z\}\times\S$, z\in\pd\H^2 is required to lie in a Lagrangian submanifold of the moduli space of flat connections over $\S$ that arises from the restrictions to the boundary of flat connections on a handle body. We establish an energy quantization result for sequences of instantons with bounded energy near $\{0\}\times\S$: Either their curvature is in fact uniformly bounded in a neighbourhood of that slice (leading to a compactness result) or there is a concentration of some minimum quantum of energy. We moreover obtain a removable singularity result for instantons with finite energy in a punctured neighbourhood of $\{0\}\times\S$. This completes the analytic foundations for the construction of an instanton Floer homology for 3-manifolds with boundary. This Floer homology is an intermediate object in the program proposed by Salamon for the proof of the Atiyah-Floer conjecture for homology-3-spheres. In the interior case, for anti-self-instantons on $\R^2\times\S$, our methods provide a new approach to the removable singularity theorem by Sibner-Sibner for codimension 2 singularities with a holonomy condition.Comment: 44 pages. Some corrections and rearrangements in section 5: Theorem 5.1 (now 5.3) was previously stated with incorrect assumption

Construction of a New Electroweak Sphaleron

We present a self-consistent ansatz for a new sphaleron in the electroweak standard model. The resulting field equations are solved numerically. This sphaleron sets the height of the energy barrier for the global SU(2) anomaly.Comment: 13 pages, Latex, NIKHEF-H/93-09 (June 1993), postscript file including figures 1 and 2 available by anonymous ftp, figure 3 available by fax (send request to [email protected]

The topology of asymptotically locally flat gravitational instantons

In this letter we demonstrate that the intersection form of the Hausel--Hunsicker--Mazzeo compactification of a four dimensional ALF gravitational instanton is definite and diagonalizable over the integers if one of the Kahler forms of the hyper-Kahler gravitational instanton metric is exact. This leads to the topological classification of these spaces. The proof exploits the relationship between L^2 cohomology and U(1) anti-instantons over gravitational instantons recognized by Hitchin. We then interprete these as reducible points in a singular SU(2) anti-instanton moduli space over the compactification leading to the identification of its intersection form. This observation on the intersection form might be a useful tool in the full geometric classification of various asymptotically locally flat gravitational instantons.Comment: 9 pages, LaTeX, no figures; Some typos corrected, slightly differs from the published versio

Spherically symmetric selfdual Yang-Mills instantons on curved backgrounds in all even dimensions

We present several different classes of selfdual Yang-Mills instantons in all even d backgrounds with Euclidean signature. In d=4p+2 the only solutions we found are on constant curvature dS and AdS backgrounds, and are evaluated in closed form. In d=4p an interesting class of instantons are given on black hole backgrounds. One class of solutions are (Euclidean) time-independent and spherically symmetric in d-1 dimensions, and the other class are spherically symmetric in all d dimensions. Some of the solutions in the former class are evaluated numerically, all the rest being given in closed form. Analytic proofs of existence covering all numerically evaluated solutions are given. All instantons studied have finite action and vanishing energy momentum tensor and do not disturb the geometry.Comment: 41 pages, 3 figure

Existence of a New Instanton in Constrained Yang-Mills-Higgs Theory

Our goal is to discover possible new 4-dimensional euclidean solutions (instantons) in fundamental SU(2) Yang-Mills-Higgs theory, with a constraint added to prevent collapse of the scale. We show that, most likely, there exists one particular new constrained instanton (\Istar) with vanishing Pontryagin index. This is based on a topological argument that involves the construction of a non-contractible loop of 4-dimensional configurations with a certain upperbound on the action, which we establish numerically. We expect \Istar to be the lowest action non-trivial solution in the vacuum sector of the theory. There also exists a related static, but unstable, solution, the new sphaleron \Sstar. Possible applications of \Istar to the electroweak interactions include the asymptotics of perturbation theory and the high-energy behaviour of the total cross-section.Comment: 32 pages, Latex, NIKHEF-H/93-02 (March 1993), postscript file including 10 figures available by anonymous ftp from nikhefh.nikhef.n

Compactness theorems of gradient Ricci solitons

In this paper, we prove the compactness theorem for gradient Ricci solitons. Let $(M_{\alpha}, g_{\alpha})$ be a sequence of compact gradient Ricci solitons of dimension $n\geq 4$, whose curvatures have uniformly bounded $L^{\frac{n}{2}}$ norms, whose Ricci curvatures are uniformly bounded from below with uniformly lower bounded volume and with uniformly upper bounded diameter, then there must exists a subsequence $(M_{\alpha}, g_{\alpha})$ converging to a compact orbifold $(M_{\infty}, g_{\infty})$ with finitly many isolated singularities, where $g_{\infty}$ is a gradient Ricci soliton metric in an orbifold sense.Comment: 21page