The existence of non-minimal solutions of the Yang-Mills-Higgs equations overR 3 with arbitrary positive coupling constant

This paper proves the existence of a non-trivial critical point of theSU(2) Yang-Mills-Higgs functional onR ³ with arbitrary positive coupling constant. The critical point lies in the zero monopole class but has action bounded strictly away from zero.Supported in part by NSF Grant DMS-9200576.Supported in part by NSF Grant DMS-9109491.     

Classification of singular Sobolev connections by their holonomy

For a connection on a principalSU(2) bundle over a base space with a codimension two singular set, a limit holonomy condition is stated. In dimension four, finite action implies that the condition is satisfied and an a priori estimate holds which classifies the singularity in terms of holonomy. If there is no holonomy, then a codimension two removable singularity theorem is obtained.Research partially supported by NSF Grant DMS-8701813Research partially supported by NSF Grant INT-8511481     

Symmetric instantons and the ADHM construction

We construct allSU(2) Yang-Mills instantons onS ⁴ that admit a certain symmetry (“quadrupole symmetry”). This is accomplished by an equivariant version of the “ADHM monad” classification of instantons. This work is part of an attempt to better understand the structure of non-self-dual Yang-Mills connections with the same symmetry. J.S. was supported by NSF Grants DMS-9106807 and DMS-9404468 Part of this work was done at the 1991 Regional Geometry Institute in Park City, Utah     

Holonomy and Skyrme's Model

In this paper we consider two generalizations of the Skyrme model. One is a variational problem for maps from a compact 3-manifold to a compact Lie group. The other is a variational problem for flat connections. We describe the path components of the configuration spaces of smooth fields for each of the variational problems. We prove that the invariants separating the path components are well-defined for (not necessarily smooth) fields with finite Skyrme energy. We prove that for every possible value of these invariants there exists a minimizer of the Skyrme functional. Throughout the paper we emphasize the importance of holonomy in the Skyrme model. Some of the results may be useful in other contexts. In particular, we define the holonomy of a distributionally flat L²_loc connection; the local developing maps for such connections need not be continuous.The first author was partially supported by NSF grant DMS-0204651.The second author was partially supported by NSF grants DMS-9970638, and DMS-0200670     

Existence of black hole solutions for the Einstein-Yang/Mills equations

This paper provides a rigorous proof of the existence of an infinite number of black hole solutions to the Einstein-Yang/Mills equations with gauge groupSU(2), for any event horizon. It is also demonstrated that the ADM mass of each solutions is finite, and that the corresponding Einstein metric tends to the associated Schwarzschild metric at a rate 1/r ², asr tends to infinity.Research supported in part by the NSF, Contract No. DMS-89-05205Research supported in part by the DE, Contract No. De-FG 02-88 EF 25065     

A largek asymptotics of Witten's invariant of Seifert manifolds

We calculate a largek asymptotic expansion of the exact surgery formula for Witten'sSU(2) invariant of some Seifert manifolds. The contributions of all flat connections are identified. An agreement with the 1-loop formula is checked. A contribution of the irreducible connections appears to contain only a finite number of terms in the asymptotic series. A 2-loop correction to the contribution of the trivial connection is found to be proportional to Casson's invariant.Work supported by NSF Grant 9009850 and R.A. Welch Foundation.     

A Poisson Structure on Compact Symmetric Spaces

We present some basic results on a natural Poisson structure on any compact symmetric space. The symplectic leaves of this structure are related to the orbits of the corresponding real semisimple group on the complex flag manifold.Acknowledgements We thank Sam Evens for many useful discussions. The first author was partially supported by NSF grant DMS-0072520. The second author was partially supported by NSF(USA) grants DMS-0105195 and DMS-0072551 and by the HHY Physical Sciences Fund at the University of Hong Kong.     

Grad ø perturbations of massless Gaussian fields

We investigate weak perturbations of the continuum massless Gaussian measure by a class of approximately local analytic functionals and use our general results to give a new proof that the pressure of the dilute dipole gas is analytic in the activity.Research partially supported by NSF Grant DMS-8802912Research partially supported by NSF Grant DMS-8601978 and DMS-8806731     

Time-Asymptotic Behavior of Wave Propagation Around a Viscous Shock Profile

We study the nonlinear stability of shock waves for viscous conservation laws. Our approach is based on a new construction of a fundamental solution for a linearized system around a shock profile. We obtain, for the first time, the pointwise estimates of nonlinear wave interactions across a shock wave. Our results apply to all ranges of weak shock waves and small perturbations. In particular, our results reduce to the time-asymptotic behavior of constant state perturbation, uniformly as the strength of the shock wave tends to zero. The research of the first author was partially supported by NSC Grant 96-2628-M-001-011 and NSF Grant DMS-0709248. The research of the second author was partially supported by NSF Grant DMS-0207154 and UAB Advance Program, sponsored by NSF.     

A Remark on Asymptotic Completeness for the Critical Nonlinear Klein-Gordon Equation

We give a short proof of asymptotic completeness and global existence for the cubic Nonlinear Klein-Gordon equation in one dimension. Our approach to dealing with the long range behavior of the asymptotic solution is by reducing it, in hyperbolic coordinates to the study of an ODE. Similar arguments extend to higher dimensions and other long range type nonlinear problems. Mathematics Subject Classifications (2000): 35L15, 74J30, 76B15 ★ Part of this work was done while H.L. was a Member of the Institute for Advanced Study, Princeton, supported by the NSF grant DMS-0111298 to the Institute. H.L. was also partially supported by the NSF Grant DMS-0200226. † Also a member of the Institute of Advanced Study, Princeton. Supported in part by NSF grant DMS-0100490.     

Bohr-sommerfeld orbits in the moduli space of flat connections and the Verlinde dimension formula

We show how the moduli space of flatSU(2) connections on a two-manifold can be quantized in the real polarization of [15], using the methods of [6]. The dimension of the quantization, given by the number of integral fibres of the polarization, matches the Verlinde formula, which is known to give the dimension of the quantization of this space in a Kähler polarization.Supported in part by MSRI under NSF Grant 85-05550Supported in part by an NSF Graduate fellowship, and by a grant-in-aid from the J. Seward Johnson Charitable TrustSupported in part by NSF Mathematical Sciences Postdoctoral Research Fellowship DMS 88-07291. Address as of January 1, 1993: Department of Mathematics, Columbia University, New York, NY 10027, USA     

The semiclassical limit forSU(2) andSO(3) gauge theory on the torus

We prove that forSU(2) andSO(3) quantum gauge theory on a torus, holonomy expectation values with respect to the Yang-Mills measure converge, asT0, to integrals with respect to a symplectic volume measure µ₀ on the moduli space of flat connections on the bundle. These moduli spaces and the symplectic structures are described explicitly.Research supported in part by LEQSF Grant RD-A-08, and NSF Grant DMS 9400961.     

Conjugacies for Tiling Dynamical Systems

We consider tiling dynamical systems and topological conjugacies between them. We prove that the criterion of being of finite type is invariant under topological conjugacy. For substitution tiling systems under rather general conditions, including the Penrose and pinwheel systems, we show that substitutions are invertible and that conjugacies are generalized sliding block codes.Research supported in part by NSF Vigre Grant DMS-0091946Research supported in part by NSF Grant DMS-0071643 and Texas ARP Grant 003658-158Acknowledgement The authors are grateful for the support of the Banff International Research Station, at which we formulated and proved Theorem 3.     

Frobenius Modules and Hodge Asymptotics

We exhibit a direct correspondence between the potential defining the H^1,1 small quantum module structure on the cohomology of a Calabi-Yau manifold and the asymptotic data of the A-model variation of Hodge structure. This is done in the abstract context of polarized variations of Hodge structure and Frobenius modules. E. Cattani was partially supported by NSF Grant DMS-0099707     

Leaf-preserving quantizations of PoissonSU(2) are not coalgebra homomorphisms

Although it has been found that some deformation quantizations of the PoissonSU(2) preserve symplectic leaves and some preserve the group (i.e. coalgebra) operation, this paper shows that a quantization ofSU(2) cannot be both leaf-preserving and group-preserving.Partially supported by NSF-Grant DMS-9002895 and DMS-9303231     

The variational problems for classical N-vector models

We prove existence and regularity properties of solutions of the variational problems introduced in the previous paper [1] for classical lattice N-vector models. These results form a basis of our renormalization group approach to low temperature expansions for the considered models.The work has been partially supported by the NSF Grant DMS-9102639.     

Quantization of the PoissonSU(2) and its Poisson homogeneous space — The 2-sphere

We show that deformation quantizations of the Poisson structures on the Poisson Lie groupSU(2) and its homogeneous space, the 2-sphere, are compatible with Woronowicz's deformation quantization ofSU(2)'s group structure and Podles' deformation quantization of 2-sphere's homogeneous structure, respectively. So in a certain sense the multiplicativity of the Lie Poisson structure onSU(2) at the classical level is preserved under quantization.With an Appendix by Jiang-Hua Lu and Alan Weinstein Department of Mathematics, University of California, Berkely, CA 94720 USAPartially supported by NSF-Grant DMS-8505550     

On a Penrose Inequality with Charge

We construct a time-symmetric asymptotically flat initial data set to the Einstein-Maxwell Equations which satisfieswhere m is the total mass, is the area radius of the outermost horizon and Q is the total charge. This yields a counter-example to a natural extension of the Penrose Inequality for charged black holes.The research of the first author was supported in part by NSF Grant DMS-0205545.The research of the second author was supported in part by NSF Grant DMS-0222387.     

Existence of infinitely-many smooth,static, global solutions of the Einstein/Yang-Mills equations

We prove the existence of infinitely-many globally defined singularity-free solutions, to the EYM equations withSU(2) gauge group. The solutions are indexed by a coupling constant, have distinct winding numbers, and their corresponding Einstein metrics decay at infinity to the flat Minkowski metric. Each solution has a finite (ADM) mass; these masses are derived from the solutions, and arenot arbitrary constants.Both authors supported in part by the NSF, Contract No. DMS 89-05205