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Steen Markvorsen

Publications and source records attributed to Steen Markvorsen.

At least 19 recordsLinked to original sources
arXiv2026

Generalized Fermat's principle and Snell's law for cone structures and applications

Fermat's principle is fully generalized to the case where a smooth interface separates two cone structures -- Lorentz-Finsler lightcones -- representing wave propagation in a potentially inhomogeneous, anisotropic, time-dependent and discontinuous medium. The interface, wave source and receiver are assumed to be a hypersurface, a submanifold and a curve in the spacetime, respectively, of any causal character. For a trajectory to fullfil Fermat's principle -- i.e., to be a critical point of the arrival time functional -- its direction must change at the interface, obeying a precise condition that generalizes Snell's law of refraction when the wave crosses the interface, or the law of reflection when it remains in a single medium. Both laws are analyzed in detail to establish the conditions ensuring the existence and uniqueness of refracted and reflected trajectories, and to determine whether they actually minimize the arrival time. Applications to Zermelo's navigation problem and the determination of geodesics in discretized spacetimes are also emphasized.

math.DG↗
arXiv2026

The convex hull of a convex space curve with four vertices

We obtain an upper bound for the volume of the convex hull of a simple closed Frenet curve with exactly four vertices, i.e., four points of vanishing torsion, and lying on the boundary of its convex hull. Moreover, we show that the upper bound is attained when the curve intersects every plane in at most four points, a condition studied by Scherk and Segre in the 1930s. The proof relies on the fact that, under the four-vertex assumption, the convex hull is a union of line segments and therefore admits an elementary parametrization. We also comment on a question posed by Newson in 1899.

math.DG↗
arXiv2025

Simultaneous Optimization of Geodesics and Fréchet Means

A central part of geometric statistics is to compute the Fréchet mean. This is a well-known intrinsic mean on a Riemannian manifold that minimizes the sum of squared Riemannian distances from the mean point to all other data points. The Fréchet mean is simple to define and generalizes the Euclidean mean, but for most manifolds even minimizing the Riemannian distance involves solving an optimization problem. Therefore, numerical computations of the Fréchet mean require solving an embedded optimization problem in each iteration. We introduce the GEORCE-FM algorithm to simultaneously compute the Fréchet mean and Riemannian distances in each iteration in a local chart, making it faster than previous methods. We extend the algorithm to Finsler manifolds and introduce an adaptive extension such that GEORCE-FM scales to a large number of data points. Theoretically, we show that GEORCE-FM has global convergence and local quadratic convergence and prove that the adaptive extension converges in expectation to the Fréchet mean. We further empirically demonstrate that GEORCE-FM outperforms existing baseline methods to estimate the Fréchet mean in terms of both accuracy and runtime.

stat.ML↗
arXiv2025

Time-dependent Zermelo navigation with tacking

We address the time- and position-dependent Zermelo navigation problem within the framework of Lorentz-Finsler geometry. Since the initial work of E. Zermelo, the task is to find the time-minimizing trajectory between two regions for a moving object whose speed profile depends on time, position and direction. We give a step-by-step review of the classical formulation of the problem, where the geometric shape generated by the velocity vectors -- the speed profile indicatrix -- is strongly convex at each point. We derive new global results for the cases where the indicatrix field is only time-dependent. In such (meso-scale realistic) cases, Zermelo navigation exhibits particularly favorable properties that have not been previously explored, making them especially appealing for both theoretical and numerical investigations. Moreover, motivated by real-world phenomena and examples, we obtain novel results for non-convex (or multi-convex) navigation, i.e. when the indicatrices fail to be convex. In this new setting -- which is not unlike the corresponding Finsler setting for Snell's law -- optimal paths may involve so-called tacking, which stems from discontinuous shifts of direction of motion. The tacking behaviour thus results in zig-zag trajectories, as observed in time-optimal sailboat navigation and surprisingly also in the flight paths of far-ranging seabirds. Finally, we provide new efficient computational algorithms and illustrate the use of them to solve the Zermelo navigation (boundary value) problem, i.e. to find numerically the time-minimizing trajectory between two fixed points in the general non-convex setting.

math.DG↗
arXiv2025

Criteria for parabolicity and hyperbolicity of conductive Riemannian manifolds

Motivated by the physics of anisotropic conductive materials we consider a linear elliptic operator $Δ_{\mathcal{W}}$ of divergence type on a Riemannian manifold $(M^{n}, g)$. The operator is determined by the metric $g$ and by a given conductivity, which is modeled by a smooth self adjoint tensor field $\mathcal{W}$ of type $(1,1)$. We establish new conditions for a conductive manifold $(M, g, \mathcal{W})$ to be $\mathcal{W}$-parabolic or $\mathcal{W}$-hyperbolic. Here, by definition, a $\mathcal{W}$-hyperbolic manifold (as opposed to a $\mathcal{W}$-parabolic manifold) admits an effective electric current $J$, i.e. a bounded potential function $u$, which is a solution to the $\mathcal{W}$-Laplace equation $Δ_{\mathcal{W}}(u) = 0$ with a finite flux of the current $J = -\mathcal{W}(\nabla u)$ to infinity. We prove a number of intrinsic conditions on $g$ and $\mathcal{W}$, that tell the type ( $\mathcal{W}$-hyperbolic or $\mathcal{W}$-parabolic) of conductive Riemannian manifolds. And we prove similar extrinsic conditions for submanifolds (involving also naturally the second fundamental form), that give the type of the submanifolds when they are endowed with the inherited conductivities from the ambient conductive space. Our results are furthermore illustrated by corresponding families of examples, which emphasize how the present setting and results generalize previous findings concerning the usual Laplacian for Riemannian manifolds (with homogeneous, constant, conductivity) as well as similar recent results for weighted manifolds and submanifolds. We also present novel examples of $\mathcal{W}$-hyperbolic manifolds where the conductivity tensor is 'extracted' from the curvature tensor of the manifold itself, such as e.g. the metric equivalents of the Einstein tensor and the Schouten tensor.

math.DG↗
arXiv2025

Score Matching Riemannian Diffusion Means

Estimating means on Riemannian manifolds is generally computationally expensive because the Riemannian distance function is not known in closed-form for most manifolds. To overcome this, we show that Riemannian diffusion means can be efficiently estimated using score matching with the gradient of Brownian motion transition densities using the same principle as in Riemannian diffusion models. Empirically, we show that this is more efficient than Monte Carlo simulation while retaining accuracy and is also applicable to learned manifolds. Our method, furthermore, extends to computing the Fréchet mean and the logarithmic map for general Riemannian manifolds. We illustrate the applicability of the estimation of diffusion mean by efficiently extending Euclidean algorithms to general Riemannian manifolds with a Riemannian $k$-means algorithm and maximum likelihood Riemannian regression.

stat.OT↗
arXiv2024

Planar pseudo-geodesics and totally umbilic submanifolds

We study totally umbilic isometric immersions between Riemannian manifolds. First, we provide a novel characterization of the totally umbilic isometric immersions with parallel normalized mean curvature vector, i.e., those having nonzero mean curvature vector and such that the unit vector in the direction of the mean curvature vector is parallel in the normal bundle. Such characterization is based on a family of curves, called planar pseudo-geodesics, representing a natural extrinsic generalization of both geodesics and Riemannian circles: being planar, their Cartan development in the tangent space is planar in the ordinary sense; being pseudo-geodesics, their geodesic and normal curvatures satisfy a linear relation. We study these curves in detail and, in particular, establish their local existence and uniqueness. Moreover, in the case of codimension-one immersions, we prove the following statement: an isometric immersion $ι\colon M \hookrightarrow Q$ is totally umbilic if and only if the extrinsic shape of every geodesic of $M$ is planar. This extends a well-known result about surfaces in $\mathbb{R}^{3}$.

math.DG↗
arXiv2023

Mean exit times from submanifolds with bounded mean curvature

We show that submanifolds with infinite mean exit time can not be isometrically and minimally immersed into cylinders, horocylinders, cones, and wedges of some product spaces. Our approach is not based on the weak maximum principle at infinity, and thus it permits us to generalize previous results concerning non-immersibility of stochastically complete submanifolds. We also produce estimates for the complete tower of moments for submanifolds with small mean curvature immersed into cylinders.

math.DG↗
arXiv2023

Reconstructing anisotropic conductivities on two-dimensional Riemannian manifolds from power densities

We consider an electrically conductive compact two-dimensional Riemannian manifold with smooth boundary. This setting defines a natural conductive Laplacian on the manifold and hence also voltage potentials, current fields and corresponding power densities arising from suitable boundary conditions. Motivated by Acousto-Electric Tomography we show that if the manifold has genus zero and the metric is known, then the anisotropic conductivity can be recovered uniquely and constructively from knowledge of a few power densities. We illustrate the procedure numerically by reconstructing an anisotropic conductivity on the catenoid, i.e. the classical genus zero minimal surface in three-space.

math.AP↗
arXiv2023

Snell's law revisited and generalized via Finsler Geometry

We study the variational problem of finding the fastest path between two points that belong to different anisotropic media, each with a prescribed speed profile and a common interface. The optimal curves are Finsler geodesics that are refracted -- broken -- as they pass through the interface, due to the discontinuity of their velocities. This "breaking" must satisfy a specific condition in terms of the Finsler metrics defined by the speed profiles, thus establishing the generalized Snell's law. In the same way, optimal paths bouncing off the interface -- without crossing into the second domain -- provide the generalized law of reflection. The classical Snell's and reflection laws are recovered in this setting when the velocities are isotropic. If one considers a wave that propagates in all directions from a given ignition point, the trajectories that globally minimize the traveltime generate the wavefront at each instant of time. We study in detail the global properties of such wavefronts in the Euclidean plane with anisotropic speed profiles. Like the individual rays, they break when they encounter the discontinuity interface. But they are also broken due to the formation of cut loci -- stemming from the self-intersection of the wavefronts -- which typically appear when they approach a high-speed profile domain from a low-speed profile.

math.DG↗
arXiv2021

Cartan ribbonization and a topological inspection

We develop the concept of Cartan ribbons together with a rolling-based method to ribbonize and approximate any given surface in space by intrinsically flat ribbons. The rolling requires that the geodesic curvature along the contact curve on the surface agrees with the geodesic curvature of the corresponding Cartan development curve. Essentially, this follows from the orientational alignment of the two co-moving Darboux frames during rolling. Using closed contact center curves we obtain closed approximating Cartan ribbons that contribute zero to the total curvature integral of the ribbonization. This paves the way for a particularly simple topological inspection -- it is reduced to the question of how the ribbons organise their edges relative to each other. The Gauss--Bonnet theorem leads to this topological inspection of the vertices. Finally, we display two examples of ribbonizations of surfaces, namely of a torus using two ribbons, and of an ellipsoid using closed curvature lines as center curves for the ribbons.

math.DG↗
arXiv2017

Geodesic sprays and frozen metrics in rheonomic Lagrange manifolds

We define systems of pre-extremals for the energy functional of regular rheonomic Lagrange manifolds and show how they induce well-defined Hamilton orthogonal nets. Such nets have applications in the modelling of e.g. wildfire spread under time- and space-dependent conditions. The time function inherited from such a Hamilton net induces in turn a time-independent Finsler metric - we call it the associated frozen metric. It is simply obtained by inserting the time function from the net into the given Lagrangean. The energy pre-extremals then become ordinary Finsler geodesics of the frozen metric and the Hamilton orthogonality property is preserved during the freeze. We compare our results with previous findings of G. W. Richards concerning his application of Huyghens' principle to establish the PDE system for Hamilton orthogonal nets in 2D Randers spaces and also concerning his explicit spray solutions for time-only dependent Randers spaces. We analyze examples of time-dependent 2D Randers spaces with simple, yet non-trivial, Zermelo data; we obtain analytic and numerical solutions to their respective energy pre-extremal equations; and we display details of the resulting (frozen) Hamilton orthogonal nets.

math.DG↗
arXiv2017

Tractors and Tractrices in Riemannian Manifolds

We generalize the notion of planar bicycle tracks -- a.k.a. one-trailer systems -- to so-called tractor/tractrix systems in general Riemannian manifolds and prove explicit expressions for the length of the ensuing tractrices and for the area of the domains that are swept out by any given tractor/tractrix system. These expressions are sensitive to the curvatures of the ambient Riemannian manifold, and we prove explicit estimates for them based on Rauch's and Toponogov's comparison theorems. Moreover, the general length shortening property of tractor/tractrix systems is used to generate geodesics in homotopy classes of curves in the ambient manifold.

math.DG↗
arXiv2014

Ends, fundamental tones, and capacities of minimal submanifolds via extrinsic comparison theory

We study the volume of extrinsic balls and the capacity of extrinsic annuli in minimal submanifolds which are properly immersed with controlled radial sectional curvatures into an ambient manifold with a pole. The key results are concerned with the comparison of those volumes and capacities with the corresponding entities in a rotationally symmetric model manifold. Using the asymptotic behavior of the volumes and capacities we then obtain upper bounds for the number of ends as well as estimates for the fundamental tone of the submanifolds in question.

math.DG↗
arXiv2013

Estimates of the first Dirichlet eigenvalue from exit time moment spectra

We compute the first Dirichlet eigenvalue of a geodesic ball in a rotationally symmetric model space in terms of the moment spectrum for the Brownian motion exit times from the ball. This expression implies an estimate as exact as you want for the first Dirichlet eigenvalue of a geodesic ball in these rotationally symmetric spaces, including the real space forms of constant curvature. As an application of the model space theory we prove lower and upper bounds for the first Dirichlet eigenvalues of extrinsic metric balls in submanifolds of ambient Riemannian spaces which have model space controlled curvatures. Moreover, from this general setting we thereby obtain new generalizations of the classical and celebrated results due to McKean and Cheung--Leung concerning the fundamental tones of Cartan-Hadamard manifolds and the fundamental tones of submanifolds with bounded mean curvature in hyperbolic spaces, respectively.

math.DG↗
arXiv2010

Comparison of exit moment spectra for extrinsic metric balls

We prove explicit upper and lower bounds for the $L^1$-moment spectra for the Brownian motion exit time from extrinsic metric balls of submanifolds $P^m$ in ambient Riemannian spaces $N^{n}$. We assume that $P$ and $N$ both have controlled radial curvatures (mean curvature and sectional curvature, respectively) as viewed from a pole in $N$. The bounds for the exit moment spectra are given in terms of the corresponding spectra for geodesic metric balls in suitably warped product model spaces. The bounds are sharp in the sense that equalities are obtained in characteristic cases. As a corollary we also obtain new intrinsic comparison results for the exit time spectra for metric balls in the ambient manifolds $N^n$ themselves.

math.DG↗
arXiv2007

Extrinsic Isoperimetric Analysis on Submanifolds with Curvatures Bounded from Below

We obtain upper bounds for the isoperimetric quotients of extrinsic balls of submanifolds in ambient spaces which have a lower bound on their radial sectional curvatures. The submanifolds are themselves only assumed to have lower bounds on the radial part of the mean curvature vector field and on the radial part of the intrinsic unit normals at the boundaries of the extrinsic spheres, respectively. In the same vein we also establish lower bounds on the mean exit time for Brownian motion in the extrinsic balls. In those cases, where we may extend our analysis to hold all the way to infinity, we apply a capacity comparison technique to obtain a sufficient condition for the submanifolds to be parabolic, i.e. a condition which will guarantee that any Brownian particle, which is free to move around in the whole submanifold, is bound to eventually revisit any given neighborhood of its starting point with probability 1. The results of this paper are in a rough sense dual to similar results obtained previously by the present authors in complementary settings where we assume that the curvatures are bounded from above.

math.DG↗