Deformations of calibrated subbundles of Euclidean spaces via twisting by special sections

We extend the "bundle constructions" of calibrated submanifolds, due to Harvey--Lawson in the special Lagrangian case, and to Ionel--Karigiannis--Min-Oo in the cases of exceptional calibrations, by "twisting" the bundles by a special (harmonic, holomorphic, parallel) section of a complementary bundle. The existence of such deformations shows that the moduli space of calibrated deformations of these "calibrated subbundles" includes deformations which destroy the linear structure of the fibre.Comment: 16 pages, no figures. Version 2: Only minor cosmetic and typographical revisions. To appear in "Annals of Global Analysis and Geometry.

Asymptotically cylindrical 7-manifolds of holonomy G_2 with applications to compact irreducible G_2-manifolds

We construct examples of exponentially asymptotically cylindrical Riemannian 7-manifolds with holonomy group equal to G_2. To our knowledge, these are the first such examples. We also obtain exponentially asymptotically cylindrical coassociative calibrated submanifolds. Finally, we apply our results to show that one of the compact G_2-manifolds constructed by Joyce by desingularisation of a flat orbifold T^7/\Gamma can be deformed to one of the compact G_2-manifolds obtainable as a generalized connected sum of two exponentially asymptotically cylindrical SU(3)-manifolds via the method given by the first author (math.DG/0012189).Comment: 36 pages; v2: corrected trivial typos; v3: some arguments corrected and improved; v4: a number of improvements on presentation, paritularly in sections 4 and 6, including an added picture

On the construction of a geometric invariant measuring the deviation from Kerr data

This article contains a detailed and rigorous proof of the construction of a geometric invariant for initial data sets for the Einstein vacuum field equations. This geometric invariant vanishes if and only if the initial data set corresponds to data for the Kerr spacetime, and thus, it characterises this type of data. The construction presented is valid for boosted and non-boosted initial data sets which are, in a sense, asymptotically Schwarzschildean. As a preliminary step to the construction of the geometric invariant, an analysis of a characterisation of the Kerr spacetime in terms of Killing spinors is carried out. A space spinor split of the (spacetime) Killing spinor equation is performed, to obtain a set of three conditions ensuring the existence of a Killing spinor of the development of the initial data set. In order to construct the geometric invariant, we introduce the notion of approximate Killing spinors. These spinors are symmetric valence 2 spinors intrinsic to the initial hypersurface and satisfy a certain second order elliptic equation ---the approximate Killing spinor equation. This equation arises as the Euler-Lagrange equation of a non-negative integral functional. This functional constitutes part of our geometric invariant ---however, the whole functional does not come from a variational principle. The asymptotic behaviour of solutions to the approximate Killing spinor equation is studied and an existence theorem is presented.Comment: 36 pages. Updated references. Technical details correcte

Shifting patterns: malaria dynamics and rainfall variability in an African highland.

The long-term patterns of malaria in the East African highlands typically involve not only a general upward trend in cases but also a dramatic increase in the size of epidemic outbreaks. The role of climate variability in driving epidemic cycles at interannual time scales remains controversial, in part because it has been seen as conflicting with the alternative explanation of purely endogenous cycles exclusively generated by the nonlinear dynamics of the disease. We analyse a long temporal record of monthly cases from 1970 to 2003 in a highland of western Kenya with both a time-series epidemiological model (time-series susceptible-infected-recovered) and a statistical approach specifically developed for non-stationary patterns. Results show that multiyear cycles of malaria outbreaks appear in the 1980s, concomitant with the timing of a regime shift in the dynamics of cases; the cycles become more pronounced in the 1990s, when the coupling between disease and rainfall is also stronger as the variance of rainfall increased at the frequencies of coupling. Disease dynamics and climate forcing play complementary and interacting roles at different temporal scales. Thus, these mechanisms should not be viewed as alternative and their interaction needs to be integrated in the development of future predictive models

Deformation Theory of Periodic Monopoles (With Singularities)

Cherkis and Kapustin (Commun Math Phys 218(2): 333–371, 2001 and Commun Math Phys 234(1):1–35, 2003) introduced periodic monopoles (with singularities), i.e. monopoles on R^2xS^1 possibly singular at a finite collection of points. In this paper we show that for generic choices of parameters the moduli spaces of periodic monopoles (with singularities) with structure group SO(3) are either empty or smooth hyperkähler manifolds. Furthermore, we prove an index theorem and therefore compute the dimension of the moduli spaces