arXiv subjects

Levin Maier

Publications and source records attributed to Levin Maier.

16 recordsLinked to original sources

arXiv2026

On Conformal Flexibility of Completeness and Minimizing Geodesics on Hilbert Manifolds

This article is part of a broader programme investigating which features of finite-dimensional Riemannian geometry persist in infinite dimensions. The Hopf--Rinow theorem fails even for Hilbert manifolds: metric and geodesic completeness need not agree, and neither property guarantees a length-minimizing geodesic between prescribed endpoints. Despite this failure, our first main result shows that the conformal class of every smooth strong Riemannian metric on a smooth separable Hilbert manifold contains a smooth strong representative that is metrically and geodesically complete and such that every two points in the same connected component are joined by a length-minimizing geodesic. By contrast, our second main result establishes a local flexibility phenomenon for metric completeness that is inherently infinite-dimensional. Given any prescribed Hilbert-norm ball, a metrically complete strong metric admits a conformal deformation which is equal to one outside that ball, preserves geodesic completeness, and destroys metric completeness.

math.DG↗

arXiv2026

On Periodic Geodesics of Half-Lie Groups

A central program in infinite-dimensional Riemannian geometry is to understand which classical finite-dimensional principles remain valid beyond finite dimensions. In this article, we study the existence of periodic geodesics on Hilbert half-Lie groups equipped with strong right-invariant Riemannian metrics. Building on recently established completeness results for this class of infinite-dimensional manifolds, we prove that every nontrivial free homotopy class contains a periodic geodesic whenever the fundamental group is nontrivial. We further establish a Lyusternik-Fet type theorem in this setting. Assuming a Palais-Smale condition modulo right translations and the existence of a nontrivial higher homotopy group, we prove the existence of a nonconstant contractible periodic geodesic. Thus, as in the finite-dimensional setting, both first and higher homotopy information continue to force the existence of periodic geodesics in infinite dimensions. In addition, we describe a reduction principle based on compact finite-dimensional totally geodesic submanifolds and apply our results to groups of Sobolev diffeomorphisms equipped with right-invariant Sobolev metrics.

math.DG↗

arXiv2026

Topics in Magnetic Geometry: Interpolation, Intersections and Integrability

This paper develops new links between contact geometry, magnetic dynamics, and symmetry in exact magnetic systems. First, we establish an interpolation property for Killing magnetic systems on contact manifolds under an additional condition. Specifically, we show that the corresponding magnetic geodesic flow interpolates smoothly between the sub-Riemannian geodesic flow on the contact distribution and the flow of the vector field associated with a primitive of the magnetic field. Second, we show that Hamiltonian group actions associated with the magnetomorphism group produce Poisson-commuting integrals of motion for the magnetic flow. Finally, we obtain new structural results on totally magnetic submanifolds, showing that fixed-point sets of magnetomorphisms and intersections of totally magnetic submanifolds are again totally magnetic. The latter two results may be viewed as extensions of classical phenomena from Riemannian geometry to magnetic geometry.

math.SG↗

arXiv2026

Information Geometry via the Q-Root Transform

In this paper, we introduce \emph{$\ell^p$-information geometry}, an infinite-dimensional framework that shares key features with the geometry of the space of probability densities $ \mathrm{Dens}(M) $ on a closed manifold, while also incorporating aspects of measure-valued information geometry. We define the \emph{$\ell^2$-probability simplex} with a noncanonical differentiable structure induced via the \emph{$q$-root transform} from an open subset of the $ \ell^q $-sphere. This choice makes the $q$-root transform an \emph{isometry} and allows us to construct the $\ell^2$- and $\ell^q$-Fisher--Rao geometries, including \emph{Amari--\v{C}encov $\alpha$-connections} and a \emph{Chern connection} in the $\ell^q$-setting. We then apply this framework to an infinite-dimensional linear optimization problem. We show that the corresponding gradient flow with respect to the $\ell^2$--Fisher--Rao metric can be solved explicitly, converges to a maximizer under a natural monotonicity assumption, and admits an interpretation as the geodesic flow of an \emph{exponential connection}. In particular, we prove that this $e$-connection is \emph{geodesically complete}. We further relate these flows to a \emph{completely integrable Hamiltonian system} through a \emph{momentum map} associated with a Hamiltonian torus action on infinite-dimensional complex projective space. Finally, inspired by the $\ell^2$-theory, we outline an analogous Fisher--Rao geometry for $ \mathrm{Dens}(M) $ on possibly noncompact Riemannian manifolds, showing that, with a suitable spherical differentiable structure, the square-root transform remains an \emph{isometry}.

math.SG↗

arXiv2026

Open problems in billiards and quantitative symplectic geometry

This document collects contributions to the Open Problem List in Billiards and Quantitative Symplectic Geometry, compiled following discussions during the workshop ``Billiards and quantitative symplectic geometry'' that took place at the University of Heidelberg on July 14--18, 2025.

math.SG↗

arXiv2026

Mathematical Foundations of Modeling ETL Process Chains

Extract-Transform-Load (ETL) processes are core components of modern data processing infrastructures. The throughput of processed data records can be adjusted by changing the amount of allocated resources, i.e.~the number of parallel processing threads for each of the three ETL phases, but also depends on stochastic variations in the per-record processing times. In chains of multiple consecutive ETL processes, the relation between allocated resources and overall throughput is further complicated, for example by the occurrence of bottlenecks affecting all subsequent ETL processes. We develop a mathematical model of ETL process chains that is accurate at the level of time-aggregated throughput and suitable for efficient simulation. The process chain is represented as a controlled discrete-time Markov process on a directed acyclic graph whose edges are individual ETL processes. We model the mean throughput as a bounded, monotone function of the number of parallel threads, to capture the diminishing benefit of allocating more threads. We furthermore introduce a Flow Balance postulate linking number of threads, mean throughput, and mean processing time. The stochastic processing times are then modeled by non-negative heavy-tailed distributions around the mean processing time. This framework provides a principled simulator for ETL networks and a foundation for learning- and control-based resource allocation.

cs.DC↗

arXiv2026

On Global Weak Solutions for the Magnetic Two-Component Hunter-Saxton System

We study the magnetic two-component Hunter-Saxton system (M2HS), which was recently derived in \cite{M24} as a magnetic geodesic equation on an infinite-dimensional configuration space. While the geometric framework and the global weak flow were outlined there, the present paper provides the analytical foundations of this construction from the PDE perspective. First, we derive an explicit solution formula in Lagrangian variables via a Riccati reduction, yielding an alternative proof of the blow-up criterion together with an explicit expression for the blow-up time. Second, we rigorously construct global conservative weak solutions by developing the analytic theory of the relaxed configuration space and the associated weak magnetic geodesic flow, thereby realizing the geometric program proposed in \cite{M24}.

math.AP↗

arXiv2025

On Ma\~n\'e's Critical Value for Tonelli Lagrangians on Half Lie-Groups

In this article, we introduce Tonelli Lagrangians on half-Lie groups equipped with a strong right-invariant Riemannian metric. These are right-invariant Lagrangians defined on the tangent bundle of a half-Lie group with quadratic growth on each fiber. The main examples of half-Lie groups are groups of $H^s$ or $C^k$ diffeomorphisms of compact manifolds. We show that the Euler--Lagrange flow exists globally. We then introduce three thresholds of the energy, called the Ma~ne critical values, and prove that under mild regularity and completeness assumptions on the half-Lie group, any two points can be connected by a global Tonelli minimizer above the lowest of these energy thresholds. Under an additional assumption on the Lagrangian, such a minimizer is a flow line of the Euler--Lagrange flow. This extends the work of Contreras from closed finite-dimensional manifolds to the infinite-dimensional context. Moreover, our results also extend the recent work of Bauer, Harms, and Michor from geodesic flows to Euler--Lagrange flows of Tonelli Lagrangians. As an application, we obtain global well-posedness of all Euler--Poincare'e equations associated with Tonelli Lagrangians on half-Lie groups equipped with strong right-invariant Riemannian metrics.

math.DG↗

arXiv2025

The Hopf--Rinow Theorem and Ma\~n\'e's Critical Value for Magnetic Geodesics on Half Lie-Groups

In this article, we investigate \emph{right-invariant magnetic systems} on half-Lie groups, which consist of a strong right-invariant Riemannian metric and a right-invariant closed two-form. The main examples are groups of $H^s$ or $C^k$ diffeomorphisms of compact manifolds. In this setting, we define \emph{Ma\~n\'e's critical value} on the universal cover for weakly exact right-invariant magnetic fields. First, we prove that the lift of the magnetic flow to the universal cover coincides with a Finsler geodesic flow for energies above this threshold. Finally, we show that for energies above Ma\~n\'e's critical value, the full Hopf--Rinow theorem holds for such magnetic systems, thereby generalizing the work of Contreras and Merry from closed finite-dimensional manifolds to this infinite-dimensional context. Our work extends the recent results of Bauer, Harms, and Michor from geodesic flows to magnetic geodesic flows.

math.SG↗

arXiv2025

The growth rate of closed prime magnetic geodesics on closed contact manifolds

In this paper, we prove that for any given closed contact manifold, there exists an infinite-dimensional space of Riemannian metrics which can be identified with the space of bundle metrics on the induced contact distribution. For each such metric, and for all energy levels, the number of embedded periodic orbits of the corresponding magnetic geodesic flow grows at least as fast as the number of geometrically distinct periodic Reeb orbits of period less than $t$. As a corollary, we deduce that for every closed 3-manifold which is not a graph manifold, there exists an open $C^1$-neighborhood of the set of nondegenerate contact forms such that for each contact form in this neighborhood, there exists an infinite-dimensional space of Riemannian metrics as above. For the corresponding magnetic systems, the number of prime closed magnetic geodesics grows at least exponentially on all energy levels. Consequently, the restriction of the magnetic geodesic flow to any energy surface has positive topological entropy.

math.SG↗

arXiv2025

On the contact type conjecture for exact magnetic systems

In this article, we answer-for a class of magnetic systems-a question now known as the contact type conjecture, whose origin trace back to the 1998 work of Contreras, Iturriaga, Paternain, and Paternain. For a broad class of magnetic systems, we explicitly construct, on any closed manifold, an infinite-dimensional space of exact magnetic systems, which we refer to as magnetic systems of strong geodesic type. For each such system, there exists at least one null-homologous embedded periodic orbit on every energy level, with negative action for energies below the strict Ma\~n\'e critical value. As a consequence, the corresponding energy surfaces are not of contact type below this threshold. Thus, for this class of systems, the contact type conjecture holds true. Moreover, for these systems, both the strict and the lowest Ma\~n\'e critical values can be computed explicitly, and they coincide whenever the aforementioned periodic magnetic geodesic is contractible, without requiring any additional assumptions on the manifold. Several remarkable multiplicity results also hold, guaranteeing arbitrarily large numbers of embedded null-homologous periodic magnetic geodesics on every energy level. We illustrate the richness of this class through two types of examples. First, on any non-aspherical manifold, there exists a dense subset of the space of Riemannian metrics such that, for each such metric, one can construct an infinite-dimensional space of exact magnetic fields yielding magnetic systems of strong geodesic type. Second, on any closed contact manifold for which the strong Weinstein conjecture holds, one can construct an infinite-dimensional space of Riemannian metrics such that, for each such metric, the magnetic system induced by the fixed contact form is of strong geodesic type.

math.SG↗

arXiv2025

On geometric hydrodynamics and infinite-dimensional magnetic systems

In this article, we combine V. Arnold's celebrated approach via the Euler-Arnold equation -- describing the geodesic flow on a Lie group equipped with a right-invariant metric \cite{Arnold66} -- with his formulation of the motion of a charged particle in a magnetic field \cite{ar61}. We introduce the \emph{magnetic Euler-Arnold equation}, which is the Eulerian form of the magnetic geodesic flow for an infinite-dimensional magnetic system on a Lie group endowed with a right-invariant metric and a right-invariant closed two-form serving as the magnetic field. As an illustration, we demonstrate that the Korteweg-de Vries equation, the generalized Camassa-Holm equation, the infinite conductivity equation, and the global quasi-geostrophic equations can all be interpreted as magnetic Euler-Arnold equations. In particular, we obtain both local and global well-posedness results for the magnetic Euler-Arnold equation associated with the global quasi-geostrophic equations.

math.SG↗

arXiv2025

Information Geometry on the $\ell^2$-Simplex via the $q$-Root Transform

In this paper, we introduce \emph{$\ell^p$-information geometry}, an infinite dimensional framework that shares key features with the geometry of the space of probability densities $ \mathrm{Dens}(M) $ on a closed manifold, while also incorporating aspects of measure-valued information geometry. We define the \emph{$\ell^2$-probability simplex} with a noncanonical differentiable structure induced via the \emph{$q$-root transform} from an open subset of the $\ell^p$-sphere. This structure renders the $q$-root map an \emph{isometry}, enabling the definition of \emph{Amari--\v{C}encov $\alpha$-connections} in this setting. We further construct \emph{gradient flows} with respect to the $\ell^2$ Fisher--Rao metric, which solve an infinite-dimensional linear optimization problem. These flows are intimately linked to an \emph{integrable Hamiltonian system} via a \emph{momentum map} arising from a Hamiltonian group action on the infinite-dimensional complex projective space.

math.SG↗

arXiv2025

On Ma\~n\'e's critical value for the two-component Hunter-Saxton system and a infnite dimensional magnetic Hopf-Rinow theorem

In this paper, we introduce a nonlinear system of partial differential equations, the magnetic two-component Hunter-Saxton system (M2HS). This system is formulated as a magnetic geodesic equation on an infinite-dimensional Lie group equipped with a right-invariant metric, the $\dot{H}^1$ -metric, which is closely related to the infinite-dimensional Fisher-Rao metric, and the derivative of an infinite-dimensional contact-type form as the magnetic field. We define Ma\~n\'e's critical value for exact magnetic systems on Hilbert manifolds in full generality and compute it explicitly for the (M2HS). Moreover, we establish an infinite-dimensional Hopf-Rinow theorem for this magnetic system, where Ma\~n\'e's critical value serves as the threshold beyond which the Hopf-Rinow theorem no longer holds. This geometric framework enables us to thoroughly analyze the blow-up behavior of solutions to the (M2HS). Using this insight, we extend solutions beyond blow-up by introducing and proving the existence of global conservative weak solutions. This extension is facilitated by extending the Madelung transform from an isometry into a magnetomorphism, embedding the magnetic system into a magnetic system on an infinite-dimensional sphere equipped with the derivative of the standard contact form as the magnetic field. Crucially, this setup can always be reduced, via a dynamical reduction theorem, to a totally magnetic three-sphere, providing a deeper understanding of the underlying dynamics.

math.AP↗

arXiv2025

The Hopf-Rinow theorem and the Ma\~n\'e critical value for magnetic geodesics on odd-dimensional spheres

The subject of this article are magnetic geodesics on odd-dimensional spheres endowed with the round metric and with the magnetic potential given by the standard contact form. We compute the Ma\~n\'e's critical value of the system and show that a value of the energy is supercritical if and only if all pairs of points on the sphere can be connected by a magnetic geodesic with that value of the energy. Our methods are explicit and rely on the description of the submanifolds invariant by the flow and of the symmetries of the system, which we define for a general magnetic system and call totally magnetic submanifolds and magnetomorphisms, respectively. We recover hereby the known fact that the system is super-integrable: the three-spheres obtained intersecting the ambient space with a complex plane are totally magnetic and each magnetic geodesic is tangent to a two-dimensional Clifford torus. In our study the integral of motion given by the angle between magnetic geodesics and the Reeb vector field plays a special role, and can be used to realize the magnetic flow as an interpolation between the sub-Riemannian geodesic flow of the contact distribution and the Reeb flow of the contact form.

math.SG↗

arXiv2024

Magnetic billiards and the Hofer-Zehnder capacity of disk tangent bundles of lens spaces

We compute the Hofer-Zehnder capacity of disk tangent bundles of certain lens spaces with respect to the round metric. Interestingly we find that the Hofer-Zehnder capacity does not see the covering, i.e. the capacity of the disk tangent bundle of the lens space coincides with the capacity of the disk tangent bundle of the 3-sphere covering it. In particular, this gives a first example, where Gromov width and Hofer-Zehnder capacity of a disk tangent bundle disagree. Techniques we use include for the lower bound magnetic billiards and for the upper bound Gromov-Witten invariants.

math.SG↗