arXiv research

New method shows pseudo-Anosov flows on graph manifolds can be simplified.

In this article we analyze totally periodic pseudo-Anosov flows in graph three manifolds. This means that in each Seifert fibered piece of the torus decomposition, the free homotopy class of regular fibers has a finite power which is also a finite power of the free homotopy class of a closed orbit of the flow. We show …

Paper defines the payback period for nonconventional cash flows using axioms.

Study null hypersurfaces in Lorentzian manifolds, proving Riemannian flow structure.

The main result presented here is that the flow associated with a riemannian metric and a non zero magnetic field on a compact oriented surface without boundary, under assumptions of hyperbolic type, cannot have the same length spectrum of topologically corresponding periodic orbits as the geodesic flow associated with…

Shows Anosov flows with genus one sections, supporting a conjecture.

This research studies end-periodic mapping tori and their hyperbolic structures.

In this paper nontrivial Killing vector fields of constant length and corresponding flows on smooth complete Riemannian manifolds are investigated. It is proved that such a flow on symmetric space is free or induced by a free isometric action of the circle $S^1$. The properties of the set of all points with finite (inf…

Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle $\mathrm{PSL}_2(\mathbb{Z})\backslash\mathrm{PSL}_2(\mathbb{R})$. The complement of any finite number of orbits is a hyperbolic $3$-manifold, which thus has a well-defined volume. We present strong nu…

The paper adapts results for Reeb flows and Hamiltonian flows, showing all orbits are closed have identical periods.

The study finds infinitely many periodic orbits that can be used to modify Anosov flows.

Study Vassiliev invariants and periodic orbits of Axiom A flows.

In this note we study the bilateral merchandise trade flows between 186 countries over the 1948-2005 period using data from the International Monetary Fund. We use Pajek to identify network structure and behavior across thresholds and over time. In particular, we focus on the evolution of trade "islands" in the a world…

The paper classifies periodic solitons in curve flows on the light-cone.

Study shows no periodic geodesics in jet space.

The space of non-singular flows on any given solenoid is shown to contain a generic subset consisting of flows that are not almost periodic. Whether this result carries over to Hamiltonian flows remains an open question.

Study shows convergence for mean curvature flow on almost minimal totally real submanifolds.

We explain how to apply techniques from integrable systems to construct $2k$-soliton homoclinic wave maps from the periodic Minkowski space $S^1\times R^1$ to a compact Lie group, and more generally to a compact symmetric space. We give a correspondence between solutions of the -1 flow equation associated to a compact …

The study finds dense orbits and absolute period leaves for complex flows.

New insights into pseudo-Anosov flows with special periodic orbits.

New proof confirms periodic orbit conjecture for Eulerisable flows.

The Arnold conjecture is proven for integers using Floer theory.

The author shows that equicontinuous geodesic flows on surfaces are periodic. A similar result for flows on 3-manifolds is also proven. The idea of the proof is to show that the return map is recurrent and therefore periodic.

Ancient flows converge fast with finite curvature and convexity.

Paper shows how to transform certain flows into R-covered ones.

Study proves a new formula for capillary hypersurfaces and shows a flow converging to a special shape.

We consider closed immersed hypersurfaces in $\R^3$ and $\R^4$ evolving by a special class of constrained surface diffusion flows. This class of constrained flows includes the classical surface diffusion flow. In this paper we present a Lifespan Theorem for these flows, which gives a positive lower bound on the time fo…

Ancient curve shortening flows have entropy and curvature bounds equivalent.

Paper solves degenerated circle pattern metric problem in spherical geometry.

The study classifies flows of finite curvature in 3D space.

We show that the properties of Lagrangian mean curvature flow are a special case of a more general phenomenon, concerning couplings between geometric flows of the ambient space and of totally real submanifolds. Both flows are driven by ambient Ricci curvature or, in the non-Kähler case, by its analogues. To this end we…

For Hamiltonian flows we establish the existence of periodic orbits on a sequence of level sets approaching a Bott-nondegenerate symplectic extremum of the Hamiltonian. As a consequence, we show that a charge on a compact manifold with a nondegenerate (i.e. symplectic) magnetic field has periodic orbits on a sequence o…

The paper studies curves in Riemannian manifolds using total variation flow.

The paper generalizes index theory for periodic manifolds and proves equivalence of signatures for tori.

The main result of this article is that if a $3$-manifold $M$ supports an Anosov flow, then the number of conjugacy classes in the fundamental group of $M$ grows exponentially fast with the length of the shortest orbit representative, hereby answering a question raised by Plante and Thurston in 1972. In fact we show th…

The study preserves lower bounds of total scalar curvature under specific metric convergence.

We introduce a combinatorial curvature flow for PL metrics on compact triangulated 3-manifolds with boundary consisting of surfaces of negative Euler characteristic. The flow tends to find the complete hyperbolic metric with totally geodesic boundary on a manifold. Some of the basic properties of the combinatorial flow…

Contradicts claims about Poincaré complexes and homology manifolds.

We construct the solution to the periodic Cauchy problem of the Schrödinger flow on the sphere. Such construction of solutions is formulated explicitly and therefore a geometric algorithm of solving this periodic Cauchy problem follows. Theoretical and experimental results will be discussed.

The study finds sufficient conditions for Reeb flows to have genus zero global surfaces of section.

Study flat flow solutions to Mullins-Sekerka and area-preserving curvature flows on planar flat torus.

We consider a financial contract that delivers a single cash flow given by the terminal value of a cumulative gains process. The problem of modelling and pricing such an asset and associated derivatives is important, for example, in the determination of optimal insurance claims reserve policies, and in the pricing of r…

New findings on magnetic geodesic flows and periodic motions.

We construct new examples of self-translating surfaces for the mean curvature flow from a periodic configuration with finitely many grim reaper cylinders in each period. Because this work is an extension of the author's article on the desingularization of a finite family of grim reaper cylinders, we simply discuss the …

The paper introduces combinatorial Calabi flows to find hyperbolic metrics on surfaces with boundary.

Hamiltonian dynamical systems tend to have infinitely many periodic orbits. For example, for a broad class of symplectic manifolds almost all levels of a proper smooth Hamiltonian carry periodic orbits. The Hamiltonian Seifert conjecture is the existence problem for regular compact energy levels without periodic orbits…

The singly periodic genus-one helicoid was in the origin of the discovery of the first example of a complete minimal surface with finite topology but infinite total curvature, the celebrated Hoffman-Karcher-Wei's genus one helicoid. The objective of this paper is to give a uniqueness theorem for the singly periodic gen…

We survey some results on the existence (and non-existence) of periodic Reeb orbits on contact manifolds, both in the open and closed case. We place these statements in the context of Finsler geometry by including a proof of the folklore theorem that the Finsler geodesic flow can be interpreted as a Reeb flow. As a mil…