We determine the group of conformal automorphisms of the self-dual metrics on n#CP^2 due to LeBrun for n>2, and Poon for n=2. These metrics arise from an ansatz involving a circle bundle over hyperbolic three-space H^3 minus a finite number of points, called monopole points. We show that for n>2 connected sums, any conformal automorphism is a lift of an isometry of H^3 which preserves the set of monopole points. Furthermore, we prove that for n = 2, such lifts form a subgroup of index 2 in the full automorphism group, which we show is a semi-direct product (U(1) \times U(1)) \times D_4, the dihedral group of order 8.
We associate to an SU(2) hyperbolic monopole a holomorphic sphere embedded in projective space and use this to uncover various features of the monopole.
We show that many hyperbolic monopoles can be distinguished from each other via their asymptotic values in contrast to the case of Euclidean monopoles.
This thesis was motivated by a desire to understand the natural geometry of hyperbolic monopole moduli spaces. We take two approaches. Firstly we develop the twistor theory of singular hyperbolic monopoles and use it to study the geometry of their charge 1 moduli spaces. After this we introduce a new way to study the moduli spaces of both Euclidean and hyperbolic monopoles by applying Kodaira's deformation theory to the spectral curve. We obtain new results in both the Euclidean and hyperbolic cases. In particular we prove new cohomology vanishing theorems and find that the hyperbolic monopole…
We discuss the Euclidean limit of hyperbolic SU(2)-monopoles, framed at infinity, from the point of view of pluricomplex geometry. More generally, we discuss the geometry of hypercomplex manifolds arising as limits of pluricomplex manifolds.
We prove that the space of gauge equivalence classes of U(1)-invariant connections on some SU(2)-principle bundles over the 4-sphere S^4 is weakly homotopy equivalent to a component of the second loop space of the 2-sphere S^2.
We prove the conjecture that a monopole in three-dimensional anti-de Sitter space can be completely determined by its ``holographic'' image on the conformal boundary two-sphere.
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 Sob…
Motivated by strong desire to understand the natural geometry of moduli spaces of hyperbolic monopoles, we introduce and study a new type of geometry: pluricomplex geometry. It is a generalisation of hypercomplex geometry: we still have a 2-sphere of complex structures, but they no longer behave like unit imaginary quaternions. We still require, however, that the 2-sphere of complex structures determines a decomposition of the complexified tangent space as tensor product of C^{2n} and C^2. Among interesting properties of pluricomplex manifold is the existence of a canonical torsion free connec…
We study solutions to Nahm's equations with continuous symmetries and, under certain (mild) hypotheses, we classify the corresponding Ansätze. Using our classification, we construct novel Nahm data, and prescribe methods for generating further solutions. Finally, we use these results to construct new BPS monopoles with spherical symmetry.
Based on Lie group method, potential symmetry and invariant solutions for generalized quasilinear hyperbolic equations are studied. To obtain the invariant solutions in explicit form, we focus on the physically interesting situations which admit potential symmetries. Then by using the partial Lagrangian approach, we find conservation laws for this equation in three physically interesting cases.
We consider the self-dual conformal classes on n#CP^2 discovered by LeBrun. These depend upon a choice of n points in hyperbolic 3-space, called monopole points. We investigate the limiting behavior of various constant scalar curvature metrics in these conformal classes as the points approach each other, or as the points tend to the boundary of hyperbolic space. There is a close connection to the orbifold Yamabe problem, which we show is not always solvable (in contrast to the case of compact manifolds). In particular, we show that there is no constant scalar curvature orbifold metric in the c…
The curvature and the reduced curvature are basic differential invariants of the pair (Hamiltonian system, Lagrange distribution) on the symplectic manifold. It is shown that the negativity of the reduced curvature implies the hyperbolicity of any compact invariant set of the Hamiltonian flow restricted to a prescribed energy level. We consider the Hamiltonian flows of the curve of least action of natural mechanical systems in sub-Riemannian structures with symmetries. We give sufficient conditions for the reduced flows (after reduction of the first integrals induced from the symmetries) to be…
We discuss the spectral curves and rational maps associated with $SU(2)$ Bogomolny monopoles of arbitrary charge $k$. We describe the effect on the rational maps of inverting monopoles in the plane with respect to which the rational maps are defined, and discuss the monopoles invariant under such inversion. We define the strongly centred monopoles, and show they form a geodesic submanifold of the $k$-monopole moduli space. The space of strongly centred $k$-monopoles invariant under the cyclic group of rotations about a fixed axis, $C_k$, is shown to consist of several surfaces of revolution, g…
We show that the Seifert-Weber dodecahedral space $\mathsf{SW}$ is an $L$-space. The proof builds on our work relating Floer homology and spectral geometry of hyperbolic three-manifolds. A direct application of our previous techniques runs into difficulties arising from the computational complexity of the problem. We overcome this by exploiting the large symmetry group and the arithmetic and tetrahedral group structure of $\mathsf{SW}$ to prove that small eigenvalues on coexact $1$-forms must have large multiplicity.
We construct infinitely many examples of finite volume 4-manifolds with $T^3$ ends that do not admit any cusped asymptotically hyperbolic Einstein metrics yet satisfy a strict logarithmic version of the Hitchin-Thorpe inequality due to Dai-Wei. This is done by using estimates from Seiberg-Witten theory due to LeBrun as well as a method for constructing solutions to the Seiberg-Witten equations on noncompact manifolds due to Biquard. We also use constructions coming from the $Pin^-(2)$ monopole equations to obtain a larger class of manifolds where these techniques apply.
The discrete Nahm equations, a system of matrix valued difference equations, arose in the work of Braam and Austin on half-integral mass hyperbolic monopoles. We show that the discrete Nahm equations are completely integrable in a natural sense: to any solution we can associate a spectral curve and a holomorphic line-bundle over the spectral curve, such that the discrete-time DN evolution corresponds to walking in the Jacobian of the spectral curve in a straight line through the line-bundle with steps of a fixed size. Some of the implications for hyperbolic monopoles are also discussed.
I give a brief overview of the mathematical theory of Noether symmetries of multifield cosmological models, which decompose naturally into visible and Hessian (a.k.a. 'hidden') symmetries. While visible symmetries correspond to those infinitesimal isometries of the Riemannian target space of the scalar field map which preserve the scalar potential, Hessian symmetries have a much deeper theory. The latter correspond to Hesse functions, defined as solutions of the so-called Hesse equation of the target space. By definition, a Hesse manifold is a Riemannian manifold which admits nontrivial Hesse …
Starting from a real analytic surface $\mathcal{M}$ with a real analytic conformal Cartan connection A. Borówka constructed a minitwistor space of an asymptotically hyperbolic Einstein-Weyl manifold with $\mathcal{M}$ being the boundary. In this article, starting from a symmetry of conformal Cartan connection, we prove that symmetries of conformal Cartan connection on $\mathcal{M}$ can be extended to symmetries of the obtained Einstein-Weyl manifold.
