arXiv subjects

Gabriel P. Paternain

Publications and source records attributed to Gabriel P. Paternain.

At least 19 recordsLinked to original sources

arXiv2026

Zoll magnetic structures and ruled surfaces

A Zoll magnetic system on an oriented closed surface $M$ is a Riemannian metric $g$ together with a function $\lambda\colon M\to \mathbb{R}$, such that every unit speed solution of the ODE $\ddot \gamma(t)=\lambda(\gamma(t))\gamma(t)^\perp$ is periodic and the minimal period depends continuously on $\gamma$. The trivial example is given by $g$ with constant curvature $K$ and $\lambda\equiv {\rm const.}$ such that $\lambda^2+K>0$. This article exhibits non-trivial Zoll magnetic systems for every genus-for genus $\ge 2$ these are the first such examples. The approach is twistor theoretic: To a general magnetic system $(g,\lambda)$ one associates its transport twistor space $Z(g,\lambda)$, which is the unit disk bundle $DM$, equipped with a degenerate complex structure that encodes the magnetic flow. For the trivial Zoll magnetic systems explicit holomorphic blow-down maps $\beta\colon Z(g,\lambda)\to W$ into certain ruled surfaces $W\to M$ are constructed, mapping $\partial Z(g,\lambda)$ onto a Lagrangian $P\subset W$. For small Lagrangian perturbations $P'\approx P$ the procedure can be reversed and this results in a large class of (non-trivial) nearby Zoll magnetic systems.

math.DG↗

arXiv2026

Local and Global Blow Downs of Transport Twistor Space

Transport twistor spaces are degenerate complex $2$-dimensional manifolds $Z$ that complexify transport problems on Riemannian surfaces, appearing, e.g., in geometric inverse problems. This article considers maps $β\colon Z\to \mathbb{C}^2$ with a holomorphic blow-down structure that resolve the degeneracy of the complex structure and allow to gain insight into the complex geometry of $Z$. The main theorems provide global $β$-maps for constant curvature metrics and their perturbations and local $β$-maps for arbitrary metrics, thereby proving a version of the classical Newlander-Nirenberg theorem for degenerate complex structures.

math.DG↗

arXiv2026

Quasi-Fuchsian flows and the coupled vortex equations

We provide an alternative construction of the quasi-Fuchsian flows introduced by Ghys in \cite{Ghys-92}. Our approach is based on the coupled vortex equations that allows to see these flows as thermostats on the unit tangent bundle of the Blaschke metric uniquely determined by a conformal class and a holomorphic quadratic differential. We also give formulas for the marked length spectrum of a quasi-Fuchsian flow in the thermostat parametrization.

math.DS↗

arXiv2025

An inverse problem for the Standard Model of particle physics

We pose and solve an inverse problem for the classical field equations that arise in the Standard Model of particle physics. Our main result describes natural conditions on the representations, so that it is possible to recover all the fields from measurements in a small set within a causal domain in Minkowski space. These conditions are satisfied for the representations arising in the Standard Model.

math.AP↗

arXiv2025

On the interplay between the light ray and the magnetic X-ray transforms

We study the light ray transform acting on tensors on a stationary Lorentzian manifold. Our main result is injectivity up to the natural obstruction as long as the associated magnetic vector field satisfies a finite degree property with respect to the vertical Fourier decomposition on the unit tangent bundle. This is based on an explicit relationship between the geodesic vector field of the Lorentzian manifold and the magnetic vector field.

math.DG↗

arXiv2025

Resonant forms at zero for dissipative Anosov flows

We study resonant differential forms at zero for transitive Anosov flows on $3$-manifolds. We pay particular attention to the dissipative case, that is, Anosov flows that do not preserve an absolutely continuous measure. Such flows have two distinguished Sinai-Ruelle-Bowen $3$-forms, $Ω_{\text{SRB}}^{\pm}$, and the cohomology classes $[ι_{X}Ω_{\text{SRB}}^{\pm}]$ (where $X$ is the infinitesimal generator of the flow) play a key role in the determination of the space of resonant $1$-forms. When both classes vanish we associate to the flow a $\textit{helicity}$ that naturally extends the classical notion associated with null-homologous volume preserving flows. We provide a general theory that includes horocyclic invariance of resonant $1$-forms and SRB-measures as well as the local geometry of the maps $X\mapsto [ι_{X}Ω_{\text{SRB}}^{\pm}]$ near a null-homologous volume preserving flow. Next, we study several relevant classes of examples. Among these are thermostats associated with holomorphic quadratic differentials, giving rise to quasi-Fuchsian flows as introduced by Ghys. For these flows we compute explicitly all resonant $1$-forms at zero, we show that $[ι_{X}Ω_{\text{SRB}}^{\pm}]=0$ and give an explicit formula for the helicity. In addition we show that a generic time change of a quasi-Fuchsian flow is semisimple and thus the order of vanishing of the Ruelle zeta function at zero is $-χ(M)$, the same as in the geodesic flow case. In contrast, we show that if $(M,g)$ is a closed surface of negative curvature, the Gaussian thermostat driven by a (small) harmonic $1$-form has a Ruelle zeta function whose order of vanishing at zero is $-χ(M)-1$.

math.DS↗

arXiv2024

Biholomorphism Rigidity for Transport Twistor Spaces

We prove that biholomorphisms between the transport twistor spaces of simple or Anosov surfaces exhibit rigidity: they must be, up to constant rescaling and the antipodal map, the lift of an orientation preserving isometry.

math.DG↗

arXiv2024

Marked length spectrum rigidity for Anosov surfaces

Let $Σ$ be a smooth closed oriented surface of genus $\geq 2$. We prove that two metrics on $Σ$ with the same marked length spectrum and Anosov geodesic flow are isometric via an isometry isotopic to the identity. The proof combines microlocal tools with the geometry of complex curves.

math.DG↗

arXiv2024

Retrieving Yang--Mills--Higgs fields in Minkowski space from active local measurements

We show that we can retrieve a Yang--Mills potential and a Higgs field (up to gauge) from source-to-solution type data associated with the classical Yang--Mills--Higgs equations in Minkowski space $\mathbb{R}^{1+3}$. We impose natural non-degeneracy conditions on the representation for the Higgs field and on the Lie algebra of the structure group which are satisfied for the case of the Standard Model. Our approach exploits the non-linear interaction of waves generated by sources with values in the centre of the Lie algebra showing that abelian components can be used effectively to recover the Higgs field.

math.AP↗

arXiv2024

Invariant distributions and the transport twistor space of closed surfaces

The purpose of this paper is to study transport equations on the unit tangent bundle of closed oriented Riemannian surfaces and to connect these to the transport twistor space of the surface (a complex surface naturally tailored to the geodesic vector field). We show that fibrewise holomorphic distributions invariant under the geodesic flow - which play an important role in tensor tomography on surfaces - form a unital algebra, that is, multiplication of such distributions is well-defined and continuous. We also exhibit a natural bijective correspondence between fibrewise holomorphic invariant distributions and genuine holomorphic functions on twistor space with polynomial blowup on the boundary of the twistor space. Eventually, when the surface is Anosov, we classify holomorphic line bundles over twistor space which are smooth up to the boundary. As a byproduct of our analysis, we obtain a quantitative version of a result of Flaminio, asserting that invariant distributions of the geodesic flow of a positively-curved metric on the 2-sphere are determined by their zeroth and first Fourier modes.

math.DG↗

arXiv2023

The Transport Oka-Grauert Principle for Simple Surfaces

This article considers the attenuated transport equation on Riemannian surfaces in the light of a novel twistor correspondence under which matrix attenuations correspond to holomorphic vector bundles on a complex surface. The main result is a transport version of the classical Oka-Grauert principle and states that the twistor space of a simple surface supports no nontrivial holomorphic vector bundles. This solves an open problem on the existence of matrix holomorphic integrating factors on simple surfaces and is applied to give a range characterisation for the non-Abelian X-ray transform. The main theorem is proved using the inverse function theorem of Nash and Moser and the required tame estimates are obtained from recent results on the injectivity of attenuated X-ray transforms and microlocal analysis of the associated normal operators.

math.DG↗

arXiv2022

The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds

We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold $Σ$ with Betti number $b_1$, the order of vanishing of the Ruelle zeta function at zero equals $4-b_1$, while in the hyperbolic case it is equal to $4-2b_1$. This is in contrast to the 2-dimensional case where the order of vanishing is a topological invariant. The proof uses the microlocal approach to dynamical zeta functions, giving a geometric description of generalized Pollicott-Ruelle resonant differential forms at 0 in the hyperbolic case and using first variation for the perturbation. To show that the first variation is generically nonzero we introduce a new identity relating pushforwards of products of resonant and coresonant 2-forms on the sphere bundle $SΣ$ with harmonic 1-forms on $Σ$.

math.DS↗

arXiv2022

The non-Abelian X-ray transform on surfaces

This paper settles the question of injectivity of the non-Abelian X-ray transform on simple surfaces for the general linear group of invertible complex matrices. The main idea is to use a factorization theorem for Loop Groups to reduce to the setting of the unitary group, where energy methods and scalar holomorphic integrating factors can be used. We also show that our main theorem extends to cover the case of an arbitrary Lie group.

math.DG↗

arXiv2021

Carleman estimates for geodesic X-ray transforms

In this article we introduce an approach for studying the geodesic X-ray transform and related geometric inverse problems by using Carleman estimates. The main result states that on compact negatively curved manifolds (resp. nonpositively curved simple or Anosov manifolds), the geodesic vector field satisfies a Carleman estimate with logarithmic weights (resp. linear weights) on the frequency side. As a particular consequence, on negatively curved simple manifolds the geodesic X-ray transform with attenuation given by a general connection and Higgs field is invertible modulo natural obstructions. The proof is based on showing that the Pestov energy identity for the geodesic vector field completely localizes in frequency. Our approach works in all dimensions $\geq 2$, on negatively curved manifolds with or without boundary, and for tensor fields of any order.

math.AP↗

arXiv2021

Statistical guarantees for Bayesian uncertainty quantification in non-linear inverse problems with Gaussian process priors

Bayesian inference and uncertainty quantification in a general class of non-linear inverse regression models is considered. Analytic conditions on the regression model $\{\mathscr G(θ): θ\in Θ\}$ and on Gaussian process priors for $θ$ are provided such that semi-parametrically efficient inference is possible for a large class of linear functionals of $θ$. A general semi-parametric Bernstein-von Mises theorem is proved that shows that the (non-Gaussian) posterior distributions are approximated by certain Gaussian measures centred at the posterior mean. As a consequence posterior-based credible sets are valid and optimal from a frequentist point of view. The theory is illustrated with two applications with PDEs that arise in non-linear tomography problems: an elliptic inverse problem for a Schrödinger equation, and inversion of non-Abelian X-ray transforms. New analytical techniques are deployed to show that the relevant Fisher information operators are invertible between suitable function spaces

math.ST↗

arXiv2020

Vortices over Riemann surfaces and dominated splittings

We associate a flow $ϕ$ to a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $ϕ$ always admits a dominated splitting and identify special cases in which $ϕ$ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$.

math.DG↗

arXiv2020

Resonant spaces for volume preserving Anosov flows

We consider Anosov flows on closed 3-manifolds preserving a volume form $Ω$. Following Dyatlov and Zworski (2017) we study spaces of invariant distributions with values in the bundle of exterior forms whose wavefront set is contained in the dual of the unstable bundle. Our first result computes the dimension of these spaces in terms of the first Betti number of the manifold, the cohomology class $[ι_{X}Ω]$ (where $X$ is the infinitesimal generator of the flow) and the helicity. These dimensions coincide with the Pollicott-Ruelle resonance multiplicities under the assumption of $\textit{semisimplicity}$. We prove various results regarding semisimplicity on 1-forms, including an example showing that it may fail for time changes of hyperbolic geodesic flows. We also study non null-homologous deformations of contact Anosov flows and we show that there is always a splitting Pollicott-Ruelle resonance on 1-forms and that semisimplicity persists in this instance. These results have consequences for the order of vanishing at zero of the Ruelle zeta function. Finally our analysis also incorporates a flat unitary twist in both, the resonant spaces and the Ruelle zeta function.

math.DS↗

arXiv2020

A sharp stability estimate for tensor tomography in non-positive curvature

We consider the geodesic X-ray transform acting on solenoidal tensor fields on a compact simply connected manifold with strictly convex boundary and non-positive curvature. We establish a stability estimate of the form $L^2\mapsto H^{1/2}_{T}$, where the $H^{1/2}_{T}$-space is defined using the natural parametrization of geodesics as initial boundary points and incoming directions (fan-beam geometry); only tangential derivatives at the boundary are used. The proof is based on the Pestov identity with boundary term localized in frequency.

math.AP↗