arXiv subjects

Daniel Fadel

Publications and source records attributed to Daniel Fadel.

10 recordsLinked to original sources

arXiv2026

Large mass limits of $\mathrm{G}_2$ and Calabi--Yau monopoles: calibrated concentration, Higgs zeros, and abelianization

We study large mass monopoles with structure group $\mathrm{SU}(2)$ or $\mathrm{SO}(3)$ on asymptotically conical $\mathrm{G}_2$-manifolds and Calabi--Yau $3$-folds, with fixed asymptotic class. After placing the AC asymptotic theory, the variational compactness theory of Parise--Pigati--Stern, and Li's singular abelian compactness theory in a common $\Theta$-monopole framework, we prove that the mass-renormalized Yang--Mills--Higgs and intermediate energy measures converge to $8\pi\|T\|$ for a compactly supported calibrated integral codimension-three cycle $T$. This identifies the two limiting currents and shows that the variational calibration inequalities are saturated. Using this common limit as the starting point for a finer analysis, if $\mathcal S$ is the calibrated support, $\mathcal Z$ the Kuratowski upper limit of the Higgs zero sets, and $\mathcal C$ the limiting nonabelian locus, defined as the Kuratowski upper limit of Li's curvature concentration loci, then $\mathcal S\subset\mathcal Z\subset\mathcal C=\mathcal S\cup\mathcal O$, where $\mathcal O$ is precisely the obstruction to effective codimension-three monotonicity. For the cohomogeneity-one large mass families on the Bryant--Salamon $\mathrm{G}_2$-manifolds and the Stenzel Calabi--Yau $3$-fold, we prove that $\mathcal O=\varnothing$. On $X\setminus\mathcal C$ the sequence abelianizes; corrected longitudinal curvatures converge smoothly, and the remaining compactness alternatives are governed by $L^2$-harmonic $2$-forms.

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arXiv2026

The large mass limit of monopoles: abelian limits and Dirac singularities

Let $(A_i,\Phi_i)$ be finite energy $\mathrm{SU}(2)$ monopoles of charge $k>0$ on an asymptotically conical $3$-manifold with one end, with masses $m_i\to\infty$. After passing to a subsequence, the mass-renormalized energy measures concentrate at finitely many points $x_a$ with concentration weights $4\pi K_a$, where $K_a$ is the total charge of the complete finite cluster of mass-one Euclidean monopoles lying over $x_a$. We prove that, on the complement $M$ of these points, the fields abelianize exponentially. After translating the Higgs fields by their masses along the unit Higgs directions and applying gauge transformations, the translated pairs converge smoothly locally to a reducible monopole $(A_\infty,\Phi_\infty)$ of the form \[ \Phi_\infty=-u\Psi_\infty, \qquad F_{A_\infty}=-*du\,\Psi_\infty, \qquad u=4\pi\sum_aK_aG(\,\cdot\,,x_a), \] where $\Psi_\infty$ is a parallel unit section and $G$ is the minimal positive Green function. Consequently, $x_a$ is a Dirac singularity of charge $K_a$. The singular part of the residual limit is determined by the weighted $0$-cycle of concentration points and total cluster charges, and does not retain the individual Euclidean profiles or their separation hierarchy. We also show that $k-\sum_aK_a$ is exactly the charge escaping through the asymptotically conical end, and describe the residual flat abelian ambiguity.

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arXiv2026

Flows of geometric structures II

We advance the general theory of flows of tensorial $\mathrm{H}$-structures, focusing on non-isometric flows and on the case $\mathrm{H}=\mathrm{SU}(m)\subset\mathrm{SO}(2m)$. After developing the relevant $\mathrm{SU}(m)$ algebra, we compare two natural evolutions: the unrestricted negative gradient flow of the intrinsic-torsion energy and a Ricci-harmonic flow. We prove short-time existence and uniqueness for the Ricci-harmonic $\mathrm{H}$-flow, with arbitrary lower-order torsion-quadratic terms, for every closed subgroup $\mathrm{H}\subset\mathrm{SO}(n)$. For groups for which the projection to $\mathfrak{h}^\perp$ defines a $4$-form, including $\{1\}$, $\mathrm{SU}(2)$, $\mathrm{G}_2$, and $\mathrm{Spin}(7)$, we express the negative gradient flow in Ricci-harmonic form up to explicit lower-order torsion terms and prove short-time existence and uniqueness by a modified DeTurck argument. We treat the genuinely different $\mathrm{SU}(m)$ case by a separate principal-symbol computation, proving short-time existence and uniqueness for the unrestricted negative gradient flow of $\mathrm{SU}(m)$-structures. The same computation identifies the natural negative gradient flow of $\mathrm{U}(m)$-structures as a borderline case, which cannot be made strictly parabolic by first-order diffeomorphism gauges. For the modified Ricci-harmonic flow, we derive heat-type evolution equations for the intrinsic torsion, a doubling-time estimate and Shi-type derivative estimates for $(|\mathrm{Rm}|^2+|\nabla T|^2+|T|^4)^{1/2}$, and a finite-time continuation criterion. In dimension six, we translate the formalism into the standard torsion forms of an $\mathrm{SU}(3)$-structure and describe, to highest order, the corresponding family of second-order quasilinear $\mathrm{SU}(3)$-flows.

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arXiv2026

Topology of isometric classes and flows of geometric structures

We revisit flows of tensorial $H$-structures for closed and connected Lie subgroups $H\leqslant\mathrm{SO}(n)$, focusing on the topology of isometric classes. We prove that the natural map assigning to an $H$-structure its induced Riemannian metric is surjective and satisfies a parametric homotopy lifting property. Since the space of Riemannian metrics is contractible, the full space of $H$-structures is homotopy equivalent to any fixed isometric class. For parallelizable manifolds, especially flat tori, these classes reduce to mapping spaces into $\mathrm{SO}(n)/H$. We discuss almost Hermitian, $\mathrm{SU}(m)$, $\mathrm{G}_2$, and $\mathrm{Spin}(7)$ structures on flat tori, showing that their isometric classes and moduli modulo orientation-preserving diffeomorphisms may have infinitely many connected components. We relate this topology to the variational theory of the intrinsic torsion energy. On the unrestricted space of $H$-structures, the functional is scale-degenerate in dimensions $n>2$: its infimum is zero on every nonempty path component, and its only critical points are torsion-free structures. Inside fixed isometric classes this homothetic escape direction is absent. We reinterpret finite-time singularity formation as concentration in nontrivial isometric homotopy classes with zero energy infimum, and contrast this with cohomological classes, such as $\mathrm{U}(3)$-structures on the flat $6$-torus, which have positive lower bounds and admit smooth harmonic representatives from holomorphic maps into $\mathbb{CP}^3$. Finally, we revisit analytical aspects of our earlier work: we prove a lifting principle for metric-dependent flows, reinterpret the Ricci $H$-flow, derive a general evolution identity for isometric flows, and extend the harmonic-flow theory beyond the original structural assumptions.

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arXiv2025

The exterior derivative and the mean value equality in $\mathbb{R}^n$

This survey revisits classical results in vector calculus and analysis by exploring a generalised perspective on the exterior derivative, interpreting it as a measure of "infinitesimal flux". This viewpoint leads to a higher-dimensional analogue of the Mean Value Theorem, valid for differential $k$-forms, and provides a natural formulation of Stokes' theorem that mirrors the exact hypotheses of the Fundamental Theorem of Calculus -- without requiring full $C^1$ smoothness of the differential form. As a numerical application, we propose an algorithm for exterior differentiation in $\mathbb{R}^n$ that relies solely on black-box access to the differential form, offering a practical tool for computation without the need for mesh discretization or explicit symbolic expressions.

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arXiv2025

$SU(2)$ Yang-Mills-Higgs functional with Higgs self-interaction on $3$-manifolds

Fixing a constant $\lambda>0$, for any parameter $\varepsilon>0$ we study critical points of the Yang--Mills--Higgs energy \[ \mathcal{Y}_{\varepsilon}(\nabla,\Phi) = \int_M \varepsilon^2|F_{\nabla}|^2 + |\nabla\Phi|^2 + \frac{\lambda}{4\varepsilon^2}(1-|\Phi|^2)^2, \] defined for pairs $(\nabla,\Phi)$, where $\nabla$ is a connection on an $SU(2)$-bundle over an oriented Riemannian $3$-manifold $(M^3, g)$, and $\Phi$ a section of the associated adjoint bundle. When $M$ is closed, we use a $2$-parameter min-max construction to produce, for $\varepsilon\ll_M 1$, non-trivial critical points in the energy regime \[ 1 \lesssim_{\lambda}\varepsilon^{-1}\mathcal{Y}_{\varepsilon}(\nabla_{\varepsilon},\Phi_{\varepsilon}) \lesssim_{\lambda, M} 1. \] When $b_1(M)=0$, these critical points are irreducible: $\nabla_{\varepsilon}\Phi_{\varepsilon}\neq 0$. Next, assuming $M$ has bounded geometry (not necessarily compact), and given critical points with $\varepsilon^{-1}\mathcal{Y}_{\varepsilon}(\nabla_{\varepsilon}, \Phi_{\varepsilon})$ uniformly bounded, we show that as $\varepsilon\to 0$, the energy measures $\varepsilon^{-1}e_{\varepsilon}(\nabla_{\varepsilon}, \Phi_{\varepsilon}) vol_{g}$ converge subsequentially to \[ |h|^2 vol_g + \sum_{x \in S}\Theta(x)\delta_{x}, \] where $h$ is an $L^2$ harmonic $1$-form, $S$ a finite set and each $\Theta(x)$ equals the energy of a finite collection of $\mathcal{Y}_{1}$-critical points on $\mathbb{R}^3$. Finally, the estimates involved also lead to an energy gap for critical points on $3$-manifolds with bounded geometry. As a byproduct of our results, we deduce the existence of non-trivial $\mathcal{Y}_{1}$-critical points over $\mathbb{R}^3$ for any $\lambda>0$.

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arXiv2022

Flows of geometric structures

We develop an abstract theory of flows of geometric $H$-structures, i.e., flows of tensor fields defining $H$-reductions of the frame bundle, for a closed and connected subgroup $H\subset SO(n)$, on any connected and oriented $n$-manifold with sufficient topology to admit such structures. The first part of the article sets up a unifying theoretical framework for deformations of $H$-structures, by way of the natural infinitesimal action of $\mathrm{GL}(n,\mathbb{R})$ on tensors combined with various bundle decompositions induced by $H$-structures. We compute evolution equations for the intrinsic torsion under general flows of $H$-structures and, as applications, we obtain general Bianchi-type identities for $H$-structures, and, for closed manifolds, a general first variation formula for the $L^2$-Dirichlet energy functional $\mathcal{E}$ on the space of $H$-structures. We then specialise the theory to the negative gradient flow of $\mathcal{E}$ over isometric $H$-structures, i.e., their harmonic flow. The core result is an almost monotonocity formula along the flow for a scale-invariant localised energy, similar to the classical formulae by Chen-Struwe for the harmonic map heat flow. This yields an $\varepsilon$-regularity theorem and an energy gap result for harmonic structures, as well as long-time existence for the flow under small initial energy, relative to the $L^\infty$-norm of initial torsion, in the spirit of Chen-Ding. Moreover, below a certain energy level, the absence of a torsion-free isometric $H$-structure in the initial homotopy class imposes the formation of finite-time singularities. These seemingly contrasting statements are illustrated by examples on flat $n$-tori, so long as $[\mathbb{S}^n,SO(n)/H]$ contains more than one element and the universal cover of $SO(n)/H$ is a sphere; e.g. when $n=7$ and $H=\rm G_2$, or $n=8$ and $H=\rm Spin(7)$.

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arXiv2021

Asymptotics of finite energy monopoles on AC $3$-manifolds

We study the asymptotic behavior of finite energy $\rm{SU}(2)$ monopoles, and general critical points of the $\rm{SU}(2)$ Yang--Mills--Higgs energy, on asymptotically conical $3$-manifolds with only one end. Our main results generalize classical results due to Groisser and Taubes in the particular case of the flat $3$-dimensional Euclidean space $\mathbb{R}^3$. Indeed, we prove the integrality of the monopole number, or charge, of finite energy configurations, and derive the classical energy formula establishing monopoles as absolute minima. Moreover, we prove that the covariant derivative of the Higgs field of a critical point of the energy decays quadratically along the end, and that its transverse component with respect to the Higgs field, as well as the corresponding component of the curvature of the underlying connection, actually decay exponentially. Additionally, under the assumption of positive Gaussian curvature on the asymptotic link, we prove that the curvature of any critical point connection decays quadratically. Furthermore, we deduce that any irreducible critical point converges uniformly along the conical end to a limiting configuration at infinity consisting of a reducible Yang--Mills connection and a parallel Higgs field.

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arXiv2020

The asymptotic geometry of $\rm{G}_2$-monopoles

This article investigates the asymptotics of $\rm{G}_2$-monopoles. First, we prove that when the underlying $\rm{G}_2$-manifold is nonparabolic (i.e. admits a positive Green's function), finite intermediate energy monopoles with bounded curvature have finite mass. The second main result restricts to the case when the underlying $\rm{G}_2$-manifold is asymptotically conical. In this situation, we deduce sharp decay estimates and that the connection converges, along the end, to a pseudo-Hermitian--Yang--Mills connection over the asymptotic cone. Finally, our last result exhibits a Fredholm setup describing the moduli space of finite intermediate energy monopoles on an asymptotically conical $\rm{G}_2$-manifold.

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arXiv2018

The limit of large mass monopoles

In this paper, we consider finite energy $\mathrm{SU}(2)$ monopoles on an asymptotically conical, oriented Riemannian $3$-manifold with one end. The connected components of the moduli space of monopoles in this setting are labeled by an integer called the charge. We analyze sequences of monopoles with fixed charge and unbounded mass, or equivalently unbounded Yang--Mills--Higgs energy. We prove that their limiting behavior is characterized by energy concentration along a blow-up set, which is finite, and obtain effective bounds on its cardinality depending only on the charge. We also consider the zero set formed by the accumulation points of zeros of the Higgs fields and prove that the zero set and the blow-up set coincide. At each concentration point, a moving centre analysis produces a finite cluster of mass one Euclidean monopoles whose total energy equals the concentration weight in the limiting energy measure, with no loss of mass-renormalized energy in the intervening regions. For general Yang--Mills--Higgs critical points whose energies are $O(m_i)$ as $i\to\infty$, where $m_i$ are the masses of the configurations, the blow-up set is countable and contains the limiting Higgs-zero set for every compact structure group. If the structure group is $\mathrm{SU}(2)$ or $\mathrm{SO}(3)$, a moving centre and scale selection argument also produces a nontrivial mass one Euclidean Yang--Mills--Higgs critical point at every blow-up point. We explain why this last conclusion can fail in higher rank.

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