Volume 15, Issue 1

It is proved that every Bianchi surface in E 3 of class C 4 whose asymptotic lines are geodesic parallels is either a helicoid or a surface of revolution.

We explore the harmonic-Ricci flow - that is, Ricci flow coupled with harmonic map flow - both as it arises naturally in certain principal bundle constructions related to Ricci flow and as a geometric flow in its own right. We demonstrate that one natural geometric context for the flow is a special case of the locally ℝ N -invariant Ricci flow of Lott, and provide examples of gradient solitons for the flow. We prove a version of Hamilton’s compactness theorem for the flow, and then generalize it to the category of étale Riemannian groupoids. Finally, we provide a detailed example of solutions to the flow on the Lie group Nil 3 .

In his seminal paper of 1980, Mostow constructed a family of lattices in PU(2, 1), the holomorphic isometry group of complex hyperbolic 2-space. In this paper, we use a description of these lattices given by Thurston in terms of cone metrics on the sphere, which is equivalent to Deligne and Mostow’s description of them using monodromy of hypergeometric functions. We give an explicit fundamental domain for some of Mostow’s lattices, specifically those with large phase shift. Our approach follows Parker’s approach of describing Livné’s lattices in terms of cone metrics on the sphere. The content of this paper is based on Boadi’s PhD thesis.

We define monads for framed torsion-free sheaves on Hirzebruch surfaces and use them to construct moduli spaces for these objects. These moduli spaces are smooth algebraic varieties, and we show that they are fine by constructing a universal monad.

In this article we determine, for an infinite family of maps on the plane, the topology of the surface on which the minimal regular covering occurs. This infinite family includes all Archimedean tessellations

For n ≥ 1, let ξ 1 , ..., ξ n be independent, identically distributed standard normal variables. Among nonnegative real vectors u = (u 1 , . . . , u n ) of Euclidean norm 1, the quantity E∥(u 1 ξ 1 ,..., u n ξ n ∥ ∞ is maximised when u has at most two non-zero entries, and it is minimised when u is proportional to (1,...,1). Further generalisations of this result are also discussed. As a corollary, a lower bound on the mean width of a general convex body K is derived in terms of the successive inner radii of K.

In symplectic geometry a question of great importance is whether a (Lagrangian) submanifold is displaceable, that is, if it can be made disjoint from itself by a Hamiltonian isotopy.We analyze the coadjoint orbits of SU(n) and their Lagrangian submanifolds that are the fibers of the Gelfand-Tsetlin map.We use the coadjoint action to displace a large collection of these fibers. Thenwe concentrate on the case n = 3 and apply McDuff’s method of probes to show that “most” of the generic Gelfand-Tsetlin fibers are displaceable. “Most” means “all but one” in the non-monotone case, and it means “all but a 1-parameter family” in the monotone case. In the case of a non-monotone manifold of full flags we present explicitly a unique non-displaceable Lagrangian fiber (S 1 ) 3 . This fiber was already proved to be non-displaceable in [10]. Our contribution is in displacing other fibers and thus proving the uniqueness.

We introduce a measure of (central) symmetry for a convex body K in ℝ n based on the volume of the harmonic mean of K and −K (that is, the body whose gauge is the average of the gauges of K and −K), and compare it to other classical volumetric measures of symmetry. We prove sharp inequalities between the volume of the harmonic mean of K and −K and the volume of their intersection, and we show that, if K is in John’s position, then the harmonic mean of K and −K has the smallest volume exactly when K is the Euclidean ball and the largest volume exactly when K is a simplex.

A metric Lie algebra is a Lie algebra endowed with a Euclidean inner product. A subalgebra is called flat, respectively totally geodesic, if its exponential image in the corresponding Lie group with left invariant Riemannian metric is flat, respectively a totally geodesic submanifold. A non-zero vector is geodesic, if the generated one-dimensional subspace is totally geodesic. We study geodesic vectors and flat totally geodesic subalgebras in two-step nilpotent metric Lie algebras and show that their linear structure is independent of the inner product of the metric Lie algebra. We determine the geodesic vectors and the flat totally geodesic subalgebras in the two-step nilpotent metric Lie algebras of dimension ≤ 6.

We show that the toroidal Lie group G = ℂ 2 /Λ, where Λ is the lattice generated by (1, 0), (0, 1) and (τ̂, τ͂), with τ̂ ∉ ℝ, is isomorphic to the generalized Jacobian J L of the complex elliptic curve C with modulus τ̂, defined by any divisor class L ≡ (M) + (N) of C fulfilling M − N = [℘ (τ͂) : ℘´(τ͂) : 1] ∈ C. This follows from an apparently new relation between the Weierstrass sigma and elliptic functions.