# Browsing by Subject "math.DG"

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Item Open Access A circle quotient of a $G_2$ coneAcharya, Bobby Samir; Bryant, Robert L; Salamon, SimonA 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 metric, and for both the curvature and symplectic 2-forms characterizing the reduction. All these tensors are invariant by a diagonal action of $SO(3)$ on $R^6$, which can be used effectively to describe the resulting geometrical features.Item Open Access Complex monopoles I: The Haydys monopole equationNagy, Ákos; Oliveira, GonçaloWe 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 3 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 hyperk\"ahler manifold in 3-different ways, which contains the ordinary Bogomolny moduli space as a complex Lagrangian submanifold---an (ABA)-brane---with respect to any of these structures. Moreover, using a gluing construction we find an open neighborhood of the normal bundle of this submanifold which is modeled on a neighborhood of the tangent bundle to the Bogomolny moduli space. This is analogous to the case of Higgs bundles over a Riemann surface, where the (co)tangent bundle of holomorphic bundles canonically embeds into the Hitchin moduli space. These results contrast immensely with the case of finite energy Kapustin--Witten monopoles for which we show a vanishing theorem in the second paper of this series [11]. Both papers in this series are self contained and can be read independently.Item Open Access Complex monopoles II: The Kapustin--Witten monopole equationNagy, Ákos; Oliveira, GonçaloWe 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 the Kapustin-Witten equations to 3 dimensions, thus we call its solutions Kapustin-Witten monopoles. Our main result is a vanishing theorem for these monopoles showing that the only finite energy Kapustin-Witten monopoles are the ordinary Bogomolny monopoles. We prove this by analyzing the asymptotic behavior of Kapustin-Witten monopoles, and combining our results with a recent theorem of Taubes.Item Open Access Counting Yang-Mills Dyons with Index Theorems(Phys.Rev. D, 2017-06-01) Stern, M; Yi, PWe 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 Higgs fields are turned on. The number of purely magnetic bound states is again found to be consistent with the electromagnetic duality of the N=4 SU(n) theory, as expected. We also count dyons of generic electric charges, which correspond to 1/4 BPS dyons in N=4 theories and 1/2 BPS dyons in N=2 theories. Surprisingly, the degeneracy of dyons is typically much larger than would be accounted for by a single supermultiplet of appropriate angular momentum, implying many supermutiplets of the same charge and the same mass.Item Open Access Curvature homogeneous hypersurfaces in space forms(2024-04-02) Bryant, Robert; Florit, Luis; Ziller, WolfgangItem Open Access Exterior Differential Systems and Euler-Lagrange Partial Differential EquationsBryant, Robert; Griffiths, Phillip; Grossman, DanielWe 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 chapter contains an introduction of the classical Poincare-Cartan form in the context of EDS, followed by proofs of classical results, including a solution to the relevant inverse problem, Noether's theorem on symmetries and conservation laws, and several aspects of minimal hypersurfaces. In the second chapter, the equivalence problem for Poincare-Cartan forms is solved, giving the differential invariants of such a form, identifying associated geometric structures (including a family of affine hypersurfaces), and exhibiting certain "special" Euler-Lagrange equations characterized by their invariants. In the third chapter, we discuss a collection of Poincare-Cartan forms having a naturally associated conformal geometry, and exhibit the conservation laws for non-linear Poisson and wave equations that result from this. The fourth and final chapter briefly discusses additional PDE topics from this viewpoint--Euler-Lagrange PDE systems, higher order Lagrangians and conservation laws, identification of local minima for Lagrangian functionals, and Backlund transformations. No previous knowledge of exterior differential systems or of the calculus of variations is assumed.Item Open Access Flat metrics with a prescribed derived coframingBryant, Robert; Clelland, Jeanne NielsenThe 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 $M$ such that, first, $\mathrm{d}\omega^i = \Omega^i$ for $i=1,2,3$, and, second, the Riemannian metric $g=(\omega^1)^2+(\omega^2)^2+(\omega^3)^2$ be flat. We show that, in the `nonsingular case', i.e., when the three $2$-forms $\Omega^i_p$ span at least a $2$-dimensional subspace of $\Lambda^2(T^*_pM)$ and are real-analytic in some $p$-centered coordinates, this problem is always solvable on a neighborhood of $p\in M$, with the general solution $\omega$ depending on three arbitrary functions of two variables. Moreover, the characteristic variety of the generic solution $\omega$ can be taken to be a nonsingular cubic. Some singular situations are considered as well. In particular, we show that the problem is solvable locally when $\Omega^1,\Omega^2,\Omega^3$ are scalar multiples of a single 2-form that do not vanish simultaneously and satisfy a nondegeneracy condition. We also show by example that solutions may fail to exist when these conditions are not satisfied.Item Open Access From vortices to instantons on the Euclidean Schwarzschild manifold(2018-01-18) Nagy, Ákos; Oliveira, GonçaloThe 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 discovered. While soon after, in 1978, the ADHM construction gave a complete description of the moduli spaces of instantons on $\rl^4$, the case of the Euclidean Schwarzschild manifold has resisted many efforts for the past 40 years. By exploring a correspondence between the planar Abelian vortices and spherically symmetric instantons on ES, we obtain: a complete description of a connected component of the moduli space of unit energy $\SU (2)$ instantons; new examples of instantons with non-integer energy (and non-trivial holonomy at infinity); a complete classification of finite energy, spherically symmetric, $\SU (2)$ instantons. As opposed to the previously known solutions, the generic instanton coming from our construction is not invariant under the full isometry group, in particular not static. Hence disproving a conjecture of Tekin.Item Open Access Geodesic behavior for Finsler metrics of constant positive flag curvature on $S^2$(2017-11-01) Bryant, RL; Foulon, P; Ivanov, SV; Matveev, VS; Ziller, WWe 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, the length of the shortest closed geodesic is a complete invariant of the geodesic flow. We also show, in any dimension, that the geodesic flow of a Finsler metrics with constant positive flag curvature is completely integrable. Finally, we give an example of a Finsler metric on~$S^2$ with positive flag curvature such that no two closed geodesics intersect and show that this is not possible when the metric is reversible or have constant flag curvatureItem Open Access Harmonic Forms, Price Inequalities, and Benjamini-Schramm ConvergenceCerbo, Luca F Di; Stern, MarkWe 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, under a negative Ricci curvature assumption and no assumption on sign of the sectional curvature, we have a convergence result for weakly uniform discrete sequences of closed Riemannian manifolds. In the negative sectional curvature case, we are able to remove the weakly uniform discreteness assumption. This is achieved by combining a refined Thick-Thin decomposition together with a Moser iteration argument for harmonic forms on manifolds with boundary.Item Open Access Instantons on multi-Taub-NUT Spaces I: Asymptotic Form and Index Theorem(Journal of Differential Geometry, 2019-12-06) Cherkis, Sergey A; Larrain-Hubach, Andres; Stern, MarkWe 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 the first in a series of papers proving the completeness of the bow construction of instantons on multi-Taub-NUT spaces and exploring it in detail.Item Open Access Instantons on multi-Taub-NUT Spaces II: Bow ConstructionCherkis, Sergey; Larraín-Hubach, Andrés; Stern, MarkUnitary 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 ALF space as the moduli space of its particular representation -- the small representation. Any other representation of that bow gives rise to anti-self-dual connections on that ALF space. We prove that each resulting connection has finite action, i.e. it is an instanton. Moreover, we derive the asymptotic form of such a connection and compute its topological class.Item Open Access Irreducible Ginzburg-Landau fields in dimension 2(2018-01-18) Nagy, ÁGinzburg--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 of these equations. Our results hold for arbitrary compact, oriented, Riemannian 2-manifolds (for example, bounded domains in $\rl^2$, spheres, tori, etc.) with de Gennes--Neumann boundary conditions. We also prove that, for each such manifold and all positive $\alpha$ and $\beta$, the Ginzburg--Landau free energy is a Palais--Smale function on the space of gauge equivalence classes, Ginzburg--Landau fields exist for only a finite set of energy values, and the moduli space of Ginzburg--Landau fields is compact.Item Open Access Laplacian Flow for Closed $G_2$-Structures: Short Time Behavior(2011-01-11) Bryant, R; Xu, FWe 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.Item Open Access Levi-flat Minimal Hypersurfaces in Two-dimensional Complex Space Forms(Adv. Stud. Pure Math., 37, Math. Soc. Japan, Tokyo, 2002, 1--44) Bryant, Robert LThe 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 nonzero, there is a 1-parameter family of such hypersurfaces. Specifically, for each one-parameter subgroup of the isometry group of the complex space form, there is an essentially unique example that is invariant under this one-parameter subgroup. On the other hand, when the curvature of the space form is zero, i.e., when the space form is complex 2-space with its standard flat metric, there is an additional `exceptional' example that has no continuous symmetries but is invariant under a lattice of translations. Up to isometry and homothety, this is the unique example with no continuous symmetries.Item Open Access Manifold Approximation by Moving Least-Squares Projection (MMLS)(Constructive Approximation) Sober, Barak; Levin, DavidIn 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 techniques (sometimes referred to as manifold learning) assume that the scattered input data is lying on a lower dimensional manifold, thus the high dimensionality problem can be overcome by learning the lower dimensionality behavior. However, in real life applications, data is often very noisy. In this work, we propose a method to approximate $\mathcal{M}$ a $d$-dimensional $C^{m+1}$ smooth submanifold of $\mathbb{R}^n$ ($d \ll n$) based upon noisy scattered data points (i.e., a data cloud). We assume that the data points are located "near" the lower dimensional manifold and suggest a non-linear moving least-squares projection on an approximating $d$-dimensional manifold. Under some mild assumptions, the resulting approximant is shown to be infinitely smooth and of high approximation order (i.e., $O(h^{m+1})$, where $h$ is the fill distance and $m$ is the degree of the local polynomial approximation). The method presented here assumes no analytic knowledge of the approximated manifold and the approximation algorithm is linear in the large dimension $n$. Furthermore, the approximating manifold can serve as a framework to perform operations directly on the high dimensional data in a computationally efficient manner. This way, the preparatory step of dimension reduction, which induces distortions to the data, can be avoided altogether.Item Open Access New G2-holonomy cones and exotic nearly Kahler structures on S^{6}and S^{3}x S^{3}(Annals of Mathematics, 2017-01-01) Foscolo, L; Haskins, M© 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 by octonionic multiplication. Nearly Kahler 6-manifolds play a distinguished role both in the general structure theory and also because of their connection with singular spaces with holonomy group the compact exceptional Lie group G2: The metric cone over a Riemannian 6-manifold M has holonomy contained in G2 if and only if M is a nearly Kahler 6-manifold. A central problem in the field has been the absence of any complete inhomogeneous examples. We prove the existence of the first complete inhomogeneous nearly Kahler 6-manifolds by proving the existence of at least one cohomogeneity one nearly Kahler structure on the 6-sphere and on the product of a pair of 3-spheres. We conjecture that these are the only simply connected (inhomogeneous) cohomogeneity one nearly Kahler structures in six dimensions.Item Open Access Nonlinear Harmonic Forms and an Indefinite Bochner Formula(2017-06-01) Stern, MarkWe 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 results. We also introduce a variant of the Bochner formula suitable for probing the structure of the intersection form of a 4-manifold.Item Open Access Notes on exterior differential systems(2014-05-13) Bryant, RLThese 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 their use.Item Open Access Notes on Projective, Contact, and Null CurvesBryant, RobertThese 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 Klein correspondence. Most of this note consists of recounting the classical background. The main application is the explicit classification of rational null curves of low degree in Q^3. I have recently received a number of requests for these notes, so I am posting them to make them generally available.