This is the third installment in our series of articles (dg-ga/9712005, dg-ga/9710032) on the application of the PU(2) monopole equations to prove Witten's conjecture (hep-th/9411102) concerning the relation between the Donaldson and Seiberg-Witten invariants of smooth four-manifolds. The moduli space of solutions to the PU(2) monopole equations provides a noncompact cobordism between links of compact moduli spaces of U(1) monopoles of Seiberg-Witten type and the moduli space of anti-self-dual SO(3) connections, which appear as singularities in this larger moduli space. In this paper we prove the first part of a general gluing theorem for PU(2) monopoles, with the proof of the second half to appear in a companion article. The ultimate purpose of the gluing theorem is to provide topological models for neighborhoods of ideal Seiberg-Witten moduli spaces appearing in lower levels of the Uhlenbeck compactification of the moduli space of PU(2) monopoles and thus permit calculations of their contributions to Donaldson invariants using the PU(2)-monopole cobordism.
We construct periodic monopoles (with singularities), i.e. monopoles on $\mathbb{R}^{2} \times \mathbb{S}^{1}$ possibly singular at a finite collection of points, by gluing methods.
We establish a canonical gluing procedure for Seiberg-Witten monopoles on the two pieces of a closed, oriented 4-manifold X which is split along a 3-dimensional closed, oriented submanifold. We only assume that the (unperturbed) character variety is Kuranishi-smooth and the limiting maps are transversal -- then we will be able to glue regular monopoles over the irreducible points of the character variety.
In this paper we develop a new approach to the gluing problem in General Relativity, that is, the problem of matching two solutions of the Einstein equations along a spacelike or characteristic (null) hypersurface. In contrast to the previous constructions, the new perspective actively utilizes the nonlinearity of the constraint equations. As a result, we are able to remove the $10$-dimensional spaces of obstructions to the null and spacelike (asymptotically flat) gluing problems, previously known in the literature. In particular, we show that any asymptotically flat spacelike initial data can…
The present article is the first in a series whose ultimate goal is to prove the Kotschick-Morgan conjecture concerning the wall-crossing formula for the Donaldson invariants of a four-manifold with b^+ = 1. The conjecture asserts that the wall-crossing terms due to changes in the metric depend at most on the homotopy type of the four-manifold and the degree of the invariant. Our principal interest in this conjecture is due to the fact that its proof is expected to resemble that of an important intermediate step towards a proof of Witten's conjecture concerning the relation between Donaldson a…
This is the second paper in a series of papers adressing the characteristic gluing problem for the Einstein vacuum equations. We solve the codimension-$10$ characteristic gluing problem for characteristic data which are close to the Minkowski data. We derive an infinite-dimensional space of gauge-dependent charges and a $10$-dimensional space of gauge-invariant charges that are conserved by the linearized null constraint equations and act as obstructions to the gluing problem. The gauge-dependent charges can be matched by applying angular and transversal gauge transformations of the characteri…
When the rank of the bundle is $\geq 2$, in a certain sense, we found an essential obstruction for the gluing construction of $G_{2}-$instantons with $1-$dimensional singularities. It involves the Atiyah classes generated by contracting a vector in $\mathbb{C}^{3}$ with the curvature. Intuitively speaking, the gluing does not work if the tangent connection at a component of the $1-$dimensional singular locus is not the twisted Fubini-Study connection on a twisted tangent bundle of $\mathbb{P}^{2}$. Particularly, it fails if the rank of the bundle is $\geq 3$.
In the present paper, we establish a gluing construction for the Nahm pole solutions to the Kapustin-Witten equations over manifolds with boundaries and cylindrical ends. Given two Nahm pole solutions with some convergence assumptions on the cylindrical ends, we prove that there exists an obstruction class for gluing the two solutions together along the cylindrical end. In addition, we establish a local Kuranishi model for this gluing picture. As an application, we show that over any compact four-manifold with $S^3$ or $T^3$ boundary, there exists a Nahm pole solution to the obstruction pertur…
On a complete manifold, such as Euclidean 3-space or hyperbolic 3-space, the limit at infinity of the norm of the Higgs field is called the mass of the monopole. We show the existence, on hypebolic 3-space, of monopoles with given magnetic charge and arbitrary mass. Previously, aside from charge one monopoles, existence was known only for monopoles with integral mass (since these arise from U(1) invariant instantons on Euclidean 4-space). The method of proof is based on Taubes' gluing procedure, using well-separated, explicit, charge one monopoles. The analysis is carried out in a weighted Sob…
We obtain necessary and sufficient conditions for the existence of "conservation laws" on null hypersurfaces for the wave equation on general four-dimensional Lorentzian manifolds. Examples of null hypersurfaces exhibiting such conservation laws include the standard null cones of Minkowski spacetime and the degenerate horizons of extremal black holes. Another (limiting) example of such a conservation law is that which gives rise to the well-known Newman-Penrose constants along the null infinity of asymptotically flat spacetimes. The existence of such conservation laws can be viewed as an obstr…
We prove the existence of non-trivial irreducible SU(2)-monopoles with Dirac singularities on any rational homology 3-sphere, equipped with any Riemannian metric, using a gluing construction.
We prove a gluing formula for the families Seiberg-Witten invariants of families of $4$-manifolds obtained by fibrewise connected sum. Our formula expresses the families Seiberg-Witten invariants of such a connected sum family in terms of the ordinary Seiberg-Witten invariants of one of the summands, under certain assumptions on the families. We construct some variants of the families Seiberg-Witten invariants and prove the gluing formula also for these variants. One variant incorporates a twist of the families moduli space using the charge conjugation symmetry of the Seiberg-Witten equations.…
In this paper we introduce the characteristic gluing problem for the Einstein vacuum equations. We present a codimension-$10$ gluing construction for characteristic initial data which are close to the Minkowski data and we show that the $10$-dimensional obstruction space consists of gauge-invariant charges which are conserved by the linearized null constraint equations. By relating these $10$ charges to the ADM energy, linear momentum, angular momentum and the center-of-mass we prove that asymptotically flat data can be characteristically glued (including the $10$ charges) to the data of a sui…
We prove Witten's formula relating the Donaldson and Seiberg-Witten series modulo powers of degree $c+2$, with $c=-{1/4}(7χ+11σ)$, for four-manifolds obeying some mild conditions, where $χ$ and $σ$ are their Euler characteristic and signature. We use the moduli space of SO(3) monopoles as a cobordism between a link of the Donaldson moduli space of anti-self-dual SO(3) connections and links of the moduli spaces of Seiberg-Witten monopoles. Gluing techniques allow us to compute contributions from Seiberg-Witten moduli spaces lying in the first (or `one-bubble') level of the Uhlenbeck compactific…
We establish a gluing theorem for monopoles over 4--manifolds containing long necks. The theorem is stated in terms of an ungluing map defined explicitly in terms of data that appear naturally in applications. Orientations of moduli spaces are handled using Benevieri--Furi's concept of orientations of Fredholm operators of index 0.
We study complexified Bogomolny monopoles using the complex linear extension of the Hodge star operator; these monopoles can be interpreted as solutions to the Bogomolny equation with a complex gauge group. Alternatively, these equations can be obtained from dimensional reduction of the Haydys instanton equations to three dimensions, thus we call them Haydys monopoles. We find that (under mild hypotheses) the smooth locus of the moduli space of finite energy Haydys monopoles on $\mathbb{R}^3$ is a Kähler manifold containing the ordinary Bogomolny moduli space as a minimal Lagrangian submanifol…
Using quantum field-theoretic arguments, Witten has established a relation between the Donaldson and Seiberg-Witten invariants of smooth four-manifolds. In this survey article, we describe the program to prove this relation using a moduli space of PU(2) = SO(3) monopoles as a cobordism between the Donaldson moduli space of anti-self-dual SO(3) connections and moduli spaces of U(1) monopoles. We provide an overview of some of our transversality and Uhlenbeck compactness results for PU(2) monopoles, along with some of our calculations of Donaldson invariants in terms of Seiberg-Witten invariants…
We prove an analogue of the Kotschick-Morgan conjecture in the context of SO(3) monopoles, obtaining a formula relating the Donaldson and Seiberg-Witten invariants of smooth four-manifolds using the SO(3)-monopole cobordism. The main technical difficulty in the SO(3)-monopole program relating the Seiberg-Witten and Donaldson invariants has been to compute intersection pairings on links of strata of reducible SO(3) monopoles, namely the moduli spaces of Seiberg-Witten monopoles lying in lower-level strata of the Uhlenbeck compactification of the moduli space of SO(3) monopoles [arXiv:dg-ga/9710…
We give new and rather general gluing theorems for anti-self-dual (ASD) conformal structures, following the method suggested by Floer. The main result is a gluing theorem for pairs of conformally ASD manifolds `joined' across a common piece (union of connected components) of their boundaries. This theorem genuinely operates in the b-category (in the sense of Melrose) and in general the boundary of the joined manifold can be non-empty. The resulting metric is a conformally ASD b-metric or, in more traditional language, a complete conformally ASD metric with cylindrical asymptotics. We also stud…
We compute the Yamabe invariants for a new infinite class of closed $4$-dimensional manifolds by using a "twisted" version of the Seiberg-Witten equations, the $\mathrm{Pin}^-(2)$-monopole equations. The same technique also provides a new obstruction to the existence of Einstein metrics or long-time solutions of the normalised Ricci flow with uniformly bounded scalar curvature.
In this second article, we prove that any desingularization in the Gromov-Hausdorff sense of an Einstein orbifold is the result of a gluing-perturbation procedure that we develop. This builds on our first paper where we proved that a Gromov-Hausdorff convergence implied a much stronger convergence in suitable weighted Hölder spaces, in which the analysis of the present paper takes place. The description of Einstein metrics as the result of a gluing-perturbation procedure also sheds light on the local structure of the moduli space of Einstein metrics near its boundary. More importantly here, we…
This article is based on a lecture by the first author at the International Georgia Topology Conference 2001 (Athens, Georgia) and the Mathematische Arbeitstagung 2001 (Bonn, Germany). We sketch a proof of Witten's formula relating the Donaldson and Seiberg-Witten series modulo powers of degree c+2, with c = -{1/4}(7 chi + 11 sigma), for four-manifolds obeying some mild conditions, where chi and sigma are their Euler characteristic and signature. We use the moduli space of SO(3) monopoles as a cobordism between a link of the Donaldson moduli space of anti-self-dual SO(3) connections and links …
This paper circulated previously in a draft version. Now, upon general request, it is about time to distribute the more detailed (and much longer) version. The main technical issues revolve around the fine structure of the compactification of the moduli spaces of flow lines and the obstruction bundle technique, with related gluing theorems, needed in the proof of the topological invariance of the equivariant version of the Floer homology.
The use of certain critical-exponent Sobolev norms is an important feature of methods employed by Taubes to solve the anti-self-dual and similar non-linear elliptic partial differential equations. Indeed, the estimates one can obtain using these critical-exponent norms appear to be the best possible when one needs to bound the norm of a Green's operator for a Laplacian, depending on a connection varying in a non-compact family, in terms of minimal data such as the first positive eigenvalue of the Laplacian or the L^2 norm of the curvature of the connection. Following Taubes, we describe a coll…
