Browsing by Subject "math.DG"
Now showing items 1-20 of 32
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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, ... -
Instantons on multi-Taub-NUT Spaces I: Asymptotic Form and Index Theorem
(Journal of Differential Geometry, 2019-12-06)We study finite action anti-self-dual Yang-Mills connections on the multi-Taub-NUT space. We establish the curvature and the harmonic spinors decay rates and compute the index of the associated Dirac operator. This is ... -
Instantons on multi-Taub-NUT Spaces II: Bow Construction
Unitary anti-self-dual connections on Asymptotically Locally Flat (ALF) hyperk\"ahler spaces are constructed in terms of data organized in a bow. Bows generalize quivers, and the relevant bow gives rise to the underlying ... -
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 ... -
Manifold Approximation by Moving Least-Squares Projection (MMLS)
In order to avoid the curse of dimensionality, frequently encountered in Big Data analysis, there was a vast development in the field of linear and nonlinear dimension reduction techniques in recent years. These ... -
New G<inf>2</inf>-holonomy cones and exotic nearly Kahler structures on S6 and S3 x S3
(Annals of Mathematics, 2017-01-01)© 2017 Department of Mathematics, Princeton University. There is a rich theory of so-called (strict) nearly Kahler manifolds, almost-Hermitian manifolds generalising the famous almost complex structure on the 6-sphere induced ... -
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 ... -
Notes on spinors in low dimension
The purpose of these old notes (written in 1998 during a research project on holonomy of pseudo-Riemannian manifolds of type (10,1)) is to determine the orbit structure of the groups Spin(p,q) acting on their spinor spaces ...
