Roger Bielawski

Simplicial convexity and its applications

The notion of convexity was generalized by many means. Some of these notions are useful in questions of topology and analysis, for example, see [S, 12, 14-161. In this paper our aim is to give simultaneously generalizations of these concepts and to show that many classical theorems, in which the word “convex” appears remain valid by replacing this word by “simpiicial convex.” In Section 1 we define a simplicial convexity by assignation with each n elements x, ,..., x, of a topological space X and each element (A, ,..., A,) of the (n l)-dimensional simplex a formal “linear combination” @[xi ,..., x,](A, ,..., A,,)E X, which depends in a continuous way on (A, ,..., A,), but not necessarily on x, ,..., x,*. In Section 2 we give a characterization of absolute retracts in terms of simplicial convexity. In Section 3 we look for continuous selections of lower semi-continuous maps. In the last section, using the technique of Knaster-KuratowskiMazurkiewicz maps, we obtain generalizations of some theorems concerning fixed point theory and its applications [3-6, 13, 171. I am very grateful to Professor Lech Gorniewicz for his valuable remarks and aid in the preparation of this paper.

MONOPOLES AND THE GIBBONS-MANTON METRIC

Abstract:We show that, in the region where monopoles are well separated, the L2-metric on the moduli space of n-monopoles is exponentially close to the L2-invariant hyperkähler metric proposed by Gibbons and Manton. The proof is based on a description of the Gibbons–Manton metric as a metric on a certain moduli space of solutions to Nahms equations, and on twistor methods. In particular, we show how the twistor description of monopole metrics determines the asymptotic metric.nnThe construction of the Gibbons–Manton metric in terms of Nahms equations yields a class of interesting (pseudo)-hyperkähler metrics. For example we show, for each semisimple Lie group G and a maximal torus T⋚G, the existence of a G×T-invariant (pseudo)-hyperkähler manifold whose hyperkähler quotients by T are precisely Kronheimers hyperkähler metrics on G?/Tℂ. A similar result holds for Kronheimers ALE-spaces.

Asymptotic Metrics for SU(N)-Monopoles with Maximal Symmetry Breaking

Abstract:We compute the asymptotic metrics for moduli spaces of SU(N) monopoles with maximal symmetry breaking. These metrics are exponentially close to the exact monopole metric1 as soon as, for each simple root, the individual monopoles corresponding to that root are well separated. We also show that the estimates can be differentiated term by term in natural coordinates, which is a new result even for SU(2) monopoles.

Complexification and Hypercomplexification of Manifolds with a Linear Connection

We give a simple interpretation of the adapted complex structure of Lempert–Szoke and Guillemin–Stenzel: it is given by a polar decomposition of the complexified manifold. We then give a twistorial construction of an SO(3)-invariant hypercomplex structure on a neighbourhood of X in TTX, where X is a real-analytic manifold equipped with a linear connection. We show that the Nahm equations arise naturally in this context: for a connection with zero curvature and arbitrary torsion, the real sections of the twistor space can be obtained by solving Nahms equations in the Lie algebra of certain vector fields. Finally, we show that, if we start with a metric connection, then our construction yields an SO(3)-invariant hyperkahler metric.

On the hyperkhler metrics associated to singularities of nilpotent varieties

We study the hyperkähler metrics associated to minimal singularities in the nilpotent variety of a semisimple Lie group. We show that Kronheimers 4-dimensional ALE spaces are naturally realized within the context of coadjoint orbits and can be thought of as certain moduli spaces ofSU(2) invariants instantons on ℝ4∖{O} with appropriate boundary conditions.We also show that the hyperkähler metrics on the resolution of theD2 singularity arise within coadjoint orbits and that this has higher dimensional versions analogous to hyperkähler metrics onT*ℂPn. We also give an explicit description of the hyperkähler metric on the orbit of highest root vectors and, consequently, an explicit description of 3-Sasakian homogeneous metrics.

Monopoles, particles and rational functions

We prove a recent conjecture of Manton and Murray: if a polynomialp(z) of degreek — 1 is given, then anSU (2) monopole corresponding to a rational functionp(z)/q(z) with well-separated poles 1,...,k is approximately made up from charge 1 monopoles located at points (1/2 In p(i), i). We show how the rate of approximation changes with the numeratorp(z) with the result that, as long as the values of the numerator remain close together relative to the distances between poles, the above statement remains true and ceases to be so otherwise.We also show that the spectral curve of the monopole approaches the union of curves of charge 1 monopoles exponentially fast. This remains true forSU (N) monopoles.

Monopoles and Clusters

We define and study certain hyperkähler manifolds which capture the asymptotic behaviour of the SU(2)-monopole metric in regions where monopoles break down into monopoles of lower charges. The rate at which these new metrics approximate the monopole metric is exponential, as for the Gibbons-Manton metric.

Existence of closed geodesics on the moduli space of k-monopoles

We establish the existence of non-constant closed geodesics on moduli spaces of SU(2) monopoles of arbitrary charge. More generally, we show that the moduli space of strongly centred monopoles of charge k, , contains a totally geodesic submanifold which can be identified with the moduli space of strongly centred 2-monopoles for even ks and with the moduli space of centred 2-monopoles for odd ks. This submanifold consists of monopoles corresponding to k collinear equally spaced particles.

Manifolds with an Su(2)-action on the tangent bundle

We study manifolds arising as spaces of sections of complex manifolds fibering over CP 1 with the normal bundle of each section isomorphic to O(k)⊗C n

TWISTOR QUOTIENTS OF HYPERKÄHLER MANIFOLDS

We generalize the hyperkaehler quotient construction to the situation where there is no group action preserving the hyperkaehler structure but for each complex structure there is an action of a complex group preserving the corresponding complex symplectic structure. Many (known and new) hyperkaehler manifolds arise as quotients in this setting. For example, all hyperkaehler structures on semisimple coadjoint orbits of a complex semisimple Lie group