1,720,975 research outputs found
Complete noncompact Spin(7)–manifolds from self-dual Einstein 4–orbifolds
Full text linkWe present an analytic construction of complete noncompact 8–dimensional Ricciflat manifolds with holonomy Spin.7/. The construction relies on the study of the adiabatic limit of metrics with holonomy Spin.7/ on principal Seifert circle bundles over asymptotically conical G2 –orbifolds. The metrics we produce have an asymptotic geometry, so-called ALC geometry, that generalises to higher dimensions the geometry of 4–dimensional ALF hyperkähler metrics. We apply our construction to asymptotically conical G2 –metrics arising from selfdual Einstein 4–orbifolds with positive scalar curvature. As illustrative examples of the power of our construction, we produce complete noncompact Spin.7/–manifolds with arbitrarily large second Betti number and infinitely many distinct families of ALC Spin.7/–metrics on the same smooth 8–manifold
ALF gravitational instantons and collapsing Ricci-flat metrics on the <b><i>K</i>3</b> surface
Full text linkWe construct large families of new collapsing hyperkähler metrics on the K3 surface. The limit space is a flat Riemannian 3-orbifold T 3/Z 2. Away from finitely many exceptional points the collapse occurs with bounded curvature. There are at most 24 exceptional points where the curvature concentrates, which always contains the 8 fixed points of the involution on T 3. The geometry around these points is modelled by ALF gravitational instantons: of dihedral type (D k) for the fixed points of the involution on T 3 and of cyclic type (A k) otherwise. The collapsing metrics are constructed by deforming approximately hyperkähler metrics obtained by gluing ALF gravitational instantons to a background (incomplete) S 1–invariant hyperkähler metric arising from the Gibbons–Hawking ansatz over a punctured 3-torus. As an immediate application to submanifold geometry, we exhibit hyperkähler metrics on the K3 surface that admit a strictly stable minimal sphere which cannot be holomorphic with respect to any complex structure compatible with the metric. </p
Deformation theory of nearly Kähler manifolds
Full text linkNearly Kähler manifolds are the Riemannian 6-manifolds admitting real Killing spinors. Equivalently, the Riemannian cone over a nearly Kähler manifold has holonomy contained in G2. In this paper we study the deformation theory of nearly Kähler manifolds, showing that it is obstructed in general. More precisely, we show that the infinitesimal deformations of the homogeneous nearly Kähler structure on the flag manifold are all obstructed to second order
Deformation Theory of Periodic Monopoles (With Singularities)
Full text linkCherkis 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
Gravitational Instantons and Degenerations of Ricci-flat Metrics on the K3 Surface
Full text linkThe study of degenerations of metrics with special holonomy is an important theme unifying the study of convergence of Einstein metrics, the study of complete non-compact manifolds with special holonomy and the construction of spaces with special holonomy by singular perturbation methods. We survey three constructions of degenerating sequences of hyperkähler metrics on the (smooth 4-manifold underlying a complex) K3 surface—the classical Kummer construction, Gross–Wilson’s work on collapse along the fibres of an elliptic fibration, and the author’s construction of sequences collapsing to a 3-dimensional limit—describing how they fit into the general theory and highlighting the role played in each construction by gravitational instantons, i.e. complete non-compact hyperkähler 4-manifolds with decaying curvature at infinity
Infinitely many new families of complete cohomogeneity one G2-manifolds: G2 analogues of the Taub–NUT and Eguchi–Hanson spaces
Full text linkWe construct infinitely many new 1-parameter families of simply connected complete non-compact G2-manifolds with controlled geometry at infinity. The generic member of each family has so-called asymptotically locally conical (ALC) geometry. However, the nature of the asymptotic geometry changes at two special parameter values: at one special value we obtain a unique member of each family with asymptotically conical (AC) geometry; on approach to the other special parameter value the family of metrics collapses to an AC Calabi-Yau 3-fold. Our infinitely many new diffeomorphism types of AC G2-manifolds are particularly noteworthy: previously the three examples constructed by Bryant and Salamon in 1989 furnished the only known simply connected AC G2-manifolds. We also construct a closely related conically singular G2-holonomy space: away from a single isolated conical singularity, where the geometry becomes asymptotic to the G2-cone over the standard nearly Kähler structure on the product of a pair of 3-spheres, the metric is smooth and it has ALC geometry at infinity. We argue that this conically singular ALC G2-space is the natural G2 analogue of the Taub-NUT metric in 4-dimensional hyperKähler geometry and that our new AC G2-metrics are all analogues of the Eguchi-Hanson metric, the simplest ALE hyperKähler manifold. Like the Taub-NUT and Eguchi-Hanson metrics, all our examples are cohomogeneity one, i.e. they admit an isometric Lie group action whose generic orbit has codimension one
Complete non-compact G2–manifolds from asymptotically conical Calabi–Yau 3-folds
Full text linkWe develop a powerful new analytic method to construct complete non-compact G2-manifolds, i.e. Riemannian 7-manifolds (M,g) whose holonomy group is the compact exceptional Lie group G2. Our construction starts with a complete non-compact asymptotically conical Calabi-Yau 3-fold B and a circle bundle M over B satisfying a necessary topological condition. Our method then produces a 1-parameter family of circle-invariant complete G2-metrics on M that collapses to the original Calabi-Yau metric on the base B as the parameter converges to 0. The G2-metrics we construct have controlled asymptotic geometry at infinity, so-called asymptotically locally conical (ALC) metrics, and are the natural higher-dimensional analogues of the ALF metrics that are well known in 4-dimensional hyperkähler geometry. We give two illustrations of the strength of our method. Firstly we use it to construct infinitely many diffeomorphism types of complete non-compact simply connected G2-manifolds; previously only a handful of such diffeomorphism types was known. Secondly we use it to prove the existence of continuous families of complete non-compact G2-metrics of arbitrarily high dimension; previously only rigid or 1-parameter families of complete non-compact G2-metrics were known
New G2 holonomy cones and exotic nearly Kaehler structures on the 6-sphere and the product of a pair of 3-spheres
Full text linkThere is a rich theory of so-called (strict) nearly Kaehler manifolds, almost-Hermitian manifolds generalising the famous almost complex structure on the 6-sphere induced by octonionic multiplication. Nearly Kaehler 6-manifolds play a distinguished role both in the general structure theory and also because of their connection with singular spaces with holonomy group the compact exceptional Lie group G2: the metric cone over a Riemannian 6-manifold M has holonomy contained in G2 if and only if M is a nearly Kaehler 6-manifold. A central problem in the field has been the absence of any complete inhomogeneous examples. We prove the existence of the first complete inhomogeneous nearly Kaehler 6-manifolds by proving the existence of at least one cohomogeneity one nearly Kaehler structure on the 6-sphere and on the product of a pair of 3-spheres. We conjecture that these are the only simply connected (inhomogeneous) cohomogeneity one nearly Kaehler structures in six dimensions
A GLUING CONSTRUCTION FOR PERIODIC MONOPOLES
Full text linkAbstract. In [4] and [6] Cherkis and Kapustin introduced periodic monopoles (with singularities), i.e. monopoles on R2×S1 possibly singular at a finite collection of points. In [9] we proved that the moduli space of charge k periodic monopoles with n singularities is either empty or generically a smooth hyperkähler manifolds of dimension 4(k−1). In this paper we settle the existence question, constructing periodic monopoles (with singularities) by gluing methods. 1
Calorons and Constituent Monopoles
Full text linkWe study anti-self-dual Yang-Mills instantons on , also known as calorons, and their behaviour under collapse of the circle factor. In this limit, we make explicit the decomposition of calorons in terms of constituent pieces which are essentially charge monopoles. We give a gluing construction of calorons in terms of the constituents and use it to compute the dimension of the moduli space. The construction works uniformly for structure group an arbitrary compact semi-simple Lie group
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