Browsing by Subject "math.DG"
Now showing items 120 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 KapustinWitten 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 YangMills Dyons with Index Theorems
(Phys.Rev. D, 20170601)We count the supersymmetric bound states of many distinct BPS monopoles in N=4 YangMills theories and in pure N=2 YangMills theories. The novelty here is that we work in generic Coulombic vacua where more than one adjoint ... 
Exterior Differential Systems and EulerLagrange Partial Differential Equations
We use methods from exterior differential systems (EDS) to develop a geometric theory of scalar, firstorder Lagrangian functionals and their associated EulerLagrange 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
(20180118)The first irreducible solution of the $\SU (2)$ selfduality 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 $S^2$
(20171101)We study nonreversible 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 1parameter family. In particular, ... 
Harmonic Forms, Price Inequalities, and BenjaminiSchramm Convergence
We study Betti numbers of sequences of Riemannian manifolds which BenjaminiSchramm converge to their universal covers. Using the Price inequalities we developed elsewhere, we derive two distinct convergence results. First, ... 
Irreducible GinzburgLandau fields in dimension 2
(20180118)GinzburgLandau fields are the solutions of the GinzburgLandau 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
(20110111)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. 
Leviflat Minimal Hypersurfaces in Twodimensional Complex Space Forms
The purpose of this article is to classify the real hypersurfaces in complex space forms of dimension 2 that are both Leviflat 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
(20170601)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
(20140513)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 notsotypical 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 wellknown ... 
On the conformal volume of 2tori
(20150706)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 PfaffDarboux Theorem of Ekeland and Nirenberg
(20151222)The classical PfaffDarboux Theorem, which provides local `normal forms' for 1forms 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
(20170601)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 nonpositive sectional ... 
Real hypersurfaces in unimodular complex surfaces
(20040727)A unimodular complex surface is a complex 2manifold X endowed with a holomorphic volume form. A strictly pseudoconvex real hypersurface M in X inherits not only a CRstructure 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 torsionfree affine connection. In particular, it contains an account ...