The Quarterly Journal of Mathematics - current issue

MULTIPLICATIVE CHARACTER SUMS WITH TWICE-DIFFERENTIABLE FUNCTIONS

Banks, W. D., Shparlinski, I. E. — Mon, 09 Nov 2009 07:23:04 PST

For a nontrivial multiplicative character modulo p, we bound character sums taken on the integer parts of a real-valued, twice-differentiable function f whose second derivative decays at an appropriate rate. For the special case that f(x) = x with some positive real number , our bounds extend recent results of several authors.

SUMS OF OPERATOR LOGARITHMS

Clark, S. — Mon, 09 Nov 2009 07:23:04 PST

Let A and B be a pair of resolvent commuting invertible sectorial operators. We shall show that, under Kalton–Weis-type conditions, the operator log A + log B is closed and equal to log(AB).

CONTACT 5-MANIFOLDS WITH SU(2)-STRUCTURE

de Andres, L. C., Fernandez, M., Fino, A., Ugarte, L. — Mon, 09 Nov 2009 07:23:04 PST

We consider 5-manifolds with a contact form arising from a hypo structure, which we call hypo-contact. We provide existence conditions for such a structure on an oriented hypersurface of a 6-manifold with a half-flat SU(3)-structure. For half-flat manifolds with a Killing vector field X preserving the SU(3)-structure we study the geometry of the orbits space. Moreover, we describe the solvable Lie algebras admitting a hypo-contact structure. This allows us to exhibit examples of Sasakian -Einstein manifolds, as well as to prove that such structures give rise to new metrics with holonomy SU(3) and G₂.

EXAMPLES OF FREE ACTIONS ON PRODUCTS OF SPHERES

Hambleton, I., Unlu, O. — Mon, 09 Nov 2009 07:23:04 PST

We construct a non-abelian extension of S¹ by Z/3 x Z/3, and prove that acts freely and smoothly on S⁵ x S⁵. This gives new actions on S⁵ x S⁵ for an infinite family P of finite 3-groups. We also show that any finite odd-order subgroup of the exceptional Lie group G₂ admits a free smooth action on S¹¹ x S¹¹. This gives new actions on S¹¹ x S¹¹ for an infinite family E of finite groups. We explain the significance of these families P, E for the general existence problem, and correct some mistakes in the literature.

HYPERBOLIC SECTIONS IN SEIFERT-FIBERED SURFACE BUNDLES

Ichihara, K., Motegi, K. — Mon, 09 Nov 2009 07:23:04 PST

Let M be a small Seifert fiber space which has also a structure of surface bundle F x [0, 1]/{(x, 0) = (f(x), 1)} over the circle, where f: F -> F is a monodromy map with non-empty fixed point set. A typical example of such a manifold appears as the result of 0-surgery on a torus knot. For each section in M, we have a ‘projection’ in F in a natural way. We give a condition assuring that the given section in M is hyperbolic in terms of the ‘projection’ in the fiber surface. By translating the result, we give a condition to obtain pseudo-Anosov automorphisms of once punctured surfaces from a periodic automorphism.

FLOWS OF G2-STRUCTURES, I

Karigiannis, S. — Mon, 09 Nov 2009 07:23:04 PST

This is a foundational paper on flows of G₂-structures. We use local coordinates to describe the four torsion forms of a G₂-structure and derive the evolution equations for a general flow of a G₂-structure on a 7-manifold M. Specifically, we compute the evolution of the metric g, the dual 4-form and the four independent torsion forms. In the process we obtain a simple new proof of a theorem of Fernández–Gray.

As an application of our evolution equations, we derive an analogue of the second Bianchi identity in G₂-geometry which appears to be new, at least in this form. We use this result to derive explicit formulas for the Ricci tensor and part of the Riemann curvature tensor in terms of the torsion. These in turn lead to new proofs of several known results in G₂-geometry.