Now showing items 1-20 of 25

• #### A circle quotient of a $G_2$ cone ﻿

A study is made of $R^6$ as a singular quotient of the conical space $R^+\times CP^3$ with holonomy $G_2$ with respect to an obvious action by $U(1)$ on $CP^3$ with fixed points. Closed expressions are found for the induced ...
• #### Complex monopoles I: The Haydys monopole equation ﻿

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, ...
• #### Complex monopoles II: The Kapustin--Witten monopole equation ﻿

We study complexified Bogomolny monopoles in 3 dimensions by complexifying the compact structure groups. In this paper we use the conjugate linear extension of the Hodge star operator which yields a reduction of ...
• #### Counting Yang-Mills Dyons with Index Theorems ﻿

(Phys.Rev. D, 2017-06-01)
We count the supersymmetric bound states of many distinct BPS monopoles in N=4 Yang-Mills theories and in pure N=2 Yang-Mills theories. The novelty here is that we work in generic Coulombic vacua where more than one adjoint ...
• #### Exterior Differential Systems and Euler-Lagrange Partial Differential Equations ﻿

We use methods from exterior differential systems (EDS) to develop a geometric theory of scalar, first-order Lagrangian functionals and their associated Euler-Lagrange PDEs, subject to contact transformations. The first ...
• #### Flat metrics with a prescribed derived coframing ﻿

The following problem is addressed: A $3$-manifold $M$ is endowed with a triple $\Omega = (\Omega^1,\Omega^2,\Omega^3)$ of closed $2$-forms. One wants to construct a coframing $\omega= (\omega^1,\omega^2,\omega^3)$ of $M$ ...
• #### From vortices to instantons on the Euclidean Schwarzschild manifold ﻿

(2018-01-18)
The first irreducible solution of the $\SU (2)$ self-duality equations on the Euclidean Schwarzschild (ES) manifold was found by Charap and Duff in 1977, only 2 years later than the famous BPST instantons on $\rl^4$ were ...
• #### Geodesic behavior for Finsler metrics of constant positive flag curvature on $S^2$ ﻿

(2017-11-01)
We study non-reversible Finsler metrics with constant flag curvature 1 on S^2 and show that the geodesic flow of every such metric is conjugate to that of one of Katok's examples, which form a 1-parameter family. In particular, ...
• #### Harmonic Forms, Price Inequalities, and Benjamini-Schramm Convergence ﻿

We study Betti numbers of sequences of Riemannian manifolds which Benjamini-Schramm converge to their universal covers. Using the Price inequalities we developed elsewhere, we derive two distinct convergence results. First, ...
• #### Irreducible Ginzburg-Landau fields in dimension 2 ﻿

(2018-01-18)
Ginzburg--Landau fields are the solutions of the Ginzburg--Landau equations which depend on two positive parameters, $\alpha$ and $\beta$. We give conditions on $\alpha$ and $\beta$ for the existence of irreducible solutions ...
• #### Laplacian Flow for Closed $G_2$-Structures: Short Time Behavior ﻿

(2011-01-11)
We prove short time existence and uniqueness of solutions to the Laplacian flow for closed $G_2$ structures on a compact manifold $M^7$. The result was claimed in \cite{BryantG2}, but its proof has never appeared.
• #### Levi-flat Minimal Hypersurfaces in Two-dimensional Complex Space Forms ﻿

The purpose of this article is to classify the real hypersurfaces in complex space forms of dimension 2 that are both Levi-flat and minimal. The main results are as follows: When the curvature of the complex space form is ...
• #### Nonlinear Harmonic Forms and an Indefinite Bochner Formula ﻿

(2017-06-01)
We introduce the study of nonlinear harmonic forms. These are forms which minimize the $L_2$ energy in a cohomology class subject to a nonlinear constraint. In this note, we include only motivations and the most basic existence ...
• #### Notes on exterior differential systems ﻿

(2014-05-13)
These are notes for a very rapid introduction to the basics of exterior differential systems and their connection with what is now known as Lie theory, together with some typical and not-so-typical applications to illustrate ...
• #### Notes on Projective, Contact, and Null Curves ﻿

These are notes on some algebraic geometry of complex projective curves, together with an application to studying the contact curves in CP^3 and the null curves in the complex quadric Q^3 in CP^4, related by the well-known ...
• #### On the conformal volume of 2-tori ﻿

(2015-07-06)
This note provides a proof of a 1985 conjecture of Montiel and Ros about the conformal volume of tori. (This material is not really new; I'm making it available now because of requests related to recent interest ...
• #### On the convex Pfaff-Darboux Theorem of Ekeland and Nirenberg ﻿

(2015-12-22)
The classical Pfaff-Darboux Theorem, which provides local `normal forms' for 1-forms on manifolds, has applications in the theory of certain economic models. However, the normal forms needed in these models come with an ...
• #### Price Inequalities and Betti Number Growth on Manifolds without Conjugate Points ﻿

(2017-06-01)
We derive Price inequalities for harmonic forms on manifolds without conjugate points and with a negative Ricci upper bound. The techniques employed in the proof work particularly well for manifolds of non-positive sectional ...
• #### Real hypersurfaces in unimodular complex surfaces ﻿

(2004-07-27)
A unimodular complex surface is a complex 2-manifold X endowed with a holomorphic volume form. A strictly pseudoconvex real hypersurface M in X inherits not only a CR-structure but a canonical coframing as well. In this ...
• #### Recent Advances in the Theory of Holonomy ﻿

This article is a report on the status of the problem of classifying the irriducibly acting subgroups of GL(n,R) that can appear as the holonomy of a torsion-free affine connection. In particular, it contains an account ...