arXiv research

We introduce cosymplectic circles and cosymplectic spheres, which are the analogues in the cosymplectic setting of contact circles and contact spheres. We provide a complete classification of compact 3-manifolds that admit a cosymplectic circle. The properties of tautness and roundness for a cosymplectic $p$-sphere are…

Gradient flow expands curves to round shapes.

Consider an analytic map of a neighborhood of 0 in a vector space to a Euclidean space. Suppose that this map takes all germs of lines passing through 0 to germs of circles. Such a map is called rounding. We introduce a natural equivalence relation on roundings and prove that any rounding, whose differential at 0 has r…

We define a new class of knot energies (known as renormalization energies) and prove that a broad class of these energies are uniquely minimized by the round circle. Most of O'Hara's knot energies belong to this class. This proves two conjectures of O'Hara and of Freedman, He, and Wang. We also find energies not minimi…

Study shows how a curve shortens to a half-circle under specific flow.

A new curve flow preserves area and converges to a circle.

The paper studies the free elastic flow of closed curves and finds their asymptotic shape converges to a circle.

Curved loxodromes on spheres are explained and their ODE derived.

{\it Fold maps} are fundamental tools in generalizing the theory of Morse functions and its application to studies of geometric properties of manifolds. One of the fundamental and important problems in the theory of fold maps is to construct explicit fold maps, which are often difficult. In this paper, we construct new…

Study cohomology rings of 3D manifolds with round fold maps into the plane.

We show that any initial closed curve suitably close to a circle flows under length-constrained curve diffusion to a round circle in infinite time with exponential convergence. We provide an estimate on the total length of time for which such curves are not strictly convex. We further show that there are no closed tran…

The paper studies ideal flows of closed curves, classifying critical points and proving flow behavior.

Sharp chord-arc estimates for curve shortening flow on spheres.

We construct closed embedded minimal surfaces in the round three-sphere, resembling two parallel copies of the equatorial two-sphere, joined by small catenoidal bridges symmetrically arranged either along two parallel circles of the equator, or along the equatorial circle and the poles. To carry out these constructions…

We show that the infinite-dimensional space of Zoll Finsler metrics on the projective plane strongly deformation retracts to the canonical round metric. In particular, this space of Zoll Finsler metrics is connected. Moreover, the strong deformation retraction arises from a deformation of the geodesic flow of every Zol…

3D manifolds can map to a plane with specific curve patterns.

We list all analytic diffeomorphisms between an open subset of the 4-dimensional projective space and an open subset of the 4-dimensional sphere that take all line segments to arcs of round circles. These are the following: restrictions of the quaternionic Hopf fibrations and projections from a hyperplane to a sphere f…

Sharp convergence rate for curvature stability in planar free elastic flow.

Recently Andrews and Bryan [3] discovered a comparison function which allows them to shorten the classical proof of the well-known fact that the curve shortening flow shrinks embedded closed curves in the plane to a round point. Using this comparison function they estimate the length of any chord from below in terms of…

New method simplifies ideal curve flow with length constraint.

We obtain uncountably many periodic solutions to the singular Yamabe problem on a round sphere, that blow up along a great circle. These are (complete) constant scalar curvature metrics on the complement of $S^1$ inside $S^m$, $m\geq 5$, that are conformal to the round (incomplete) metric and "periodic" in the sense of…

We investigate the elastic behavior of knotted loops of springy wire. To this end we minimize the classic bending energy $E_{\text{bend}}=\intκ^2$ together with a small multiple of ropelength $\mathcal R=\text{length}/\text{thickness}$ in order to penalize selfintersection. Our main objective is to characterize elastic…

The paper bounds the min-max width of embedded circles on spheres and manifolds.

The classical isoperimetric inequality in R^3 states that the surface of smallest area enclosing a given volume is a sphere. We show that the least area surface enclosing two equal volumes is a double bubble, a surface made of two pieces of round spheres separated by a flat disk, meeting along a single circle at an ang…

For any $α>0,$ we study $k^α$-type length-preserving and area-preserving nonlocal flow of convex closed plane curves and show that these two types of flow evolve such curves into round circles in $C^{\infty}$-norm.$\$Other relevant $k^α$-type nonlocal flow is also discussed when $α\geq1.\$

For each integer $k \geq 2$, we apply gluing methods to construct sequences of minimal surfaces embedded in the round $3$-sphere. We produce two types of sequences, all desingularizing collections of intersecting Clifford tori. Sequences of the first type converge to a collection of $k$ Clifford tori intersecting with …

In earlier work of NK new closed embedded smooth minimal surfaces in the round three-sphere $\mathbb{S}^3(1)$ were constructed, each resembling two parallel copies of the equatorial two-sphere $\mathbb{S}^2_{eq}$ joined by small catenoidal bridges, with the catenoidal bridges concentrating along two parallel circles, o…

We investigate knot-theoretic properties of geometrically defined curvature energies such as integral Menger curvature. Elementary radii-functions, such as the circumradius of three points, generate a family of knot energies guaranteeing self-avoidance and a varying degree of higher regularity of finite energy curves. …

In this paper we consider the steepest descent $H^{-1}$-gradient flow of the length functional for immersed plane curves, known as the curve diffusion flow. It is known that under this flow there exist both initially immersed curves which develop at least one singularity in finite time and initially embedded curves whi…

Given a Hopf fibration of a round sphere by parallel great subspheres, we prove that the projection map to the base space is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. Similarly, given a Hopf fibration of a round sphere by parallel great circles, we view a unit…

We study invariant contact p-spheres on principal circle-bundles and solve the corresponding existence problem in dimension 3. Moreover, we show that contact p-spheres can only exist on (4n-1)-dimensional manifolds and we construct examples of contact p-spheres on such manifolds. We also consider relations between taut…

We show that uniform lattices in some semi-simple groups (notably complex ones) admit Anosov surface subgroups. This result has a quantitative version: we introduce a notion, called $K$-Sullivan maps, which generalizes the notion of $K$-quasi-circles in hyperbolic geometry, and show in particular that Sullivan maps are…

Suppose G is a non-free finitely generated Kleinian group without parabolics which is not a lattice and let C(G) denote the commensurator in PSL(2,C). We prove that if the limit set of G is not a round circle, then C(G) is discrete. Furthermore, G has finite index in C(G) unless G is a fiber group in which case C(G) is…

A fibration of a Riemannian manifold is fiberwise homogeneous if there are isometries of the manifold onto itself, taking any given fiber to any other one, and preserving fibers. Examples are fibrations of Euclidean n-space by parallel n-planes, and the Hopf fibrations of the round n-sphere by great n-spheres. In this …

For two disjoint rectifiable star-shaped Jordan curves (including round circles) in the asymptotic boundary of hyperbolic 3-space, if the distance (see Definition 1.8) between these two Jordan curves are bounded from above by some constant, then there exists an annulus-type area minimizing (or equivalently least area) …

It is known that planar disks and small spherical caps are the only constant mean curvature graphs whose boundary is a round circle. Usually, the proof invokes the Maximum Principle for elliptic equations. This paper presents a new proof of this result motivated by an article due to Reilly. Our proof utilizes a flux fo…

The paper studies a curvature flow linked to the physical phenomenon of wound closure. Under the flow we show that a closed, initially convex or close-to-convex curve shrinks to a round point in finite time. We also study the singularity, showing that the singularity profile after continuous rescaling is that of a circ…

A physically natural potential energy for simple closed curves in $\bold R^3$ is shown to be invariant under Möbius transformations. This leads to the rapid resolution of several open problems: round circles are precisely the absolute minima for energy; there is a minimum energy threshold below which knotting cannot oc…

The paper classifies flows of ancient curves in 2D space.

Eigenfunction gradients on curved spaces imply rigid structure.

Study of curve evolution in 2D space forms converging to a circle.

New insights into $p$-widths of surfaces, proving optimality and calculating constants.

The fixed-point index of a homeomorphism of Jordan curves measures the number of fixed-points, with multiplicity, of the extension of the homeomorphism to the full Jordan domains in question. The now-classical Circle Index Lemma says that the fixed-point index of a positive-orientation-preserving homeomorphism of round…

The recent proof by Bigelow and Krammer that the braid groups are linear opens the possibility of applications to the study of knots and links. It was proved by the first author and Menasco that any closed braid representative of the unknot can be systematically simplified to a round planar circle by a sequence of exch…

The paper connects complex contact structures to specific types of almost contact 3-structures.

We show that for a Schrödinger operator with bounded potential on a manifold with cylindrical ends the space of solutions which grows at most exponentially at infinity is finite dimensional and, for a dense set of potentials (or, equivalently for a surface, for a fixed potential and a dense set of metrics), the constan…

We show how to construct broken, achiral Lefschetz fibrations on arbitrary smooth, closed, oriented 4-manifolds. These are generalizations of Lefschetz fibrations over the 2-sphere, where we allow Lefschetz singularities with the non-standard orientation as well as circles of singularities corresponding to round 1-hand…

Benguria and Loss have conjectured that, amongst all smooth closed curves of length $2π$ in the plane, the lowest possible eigenvalue of the operator $L=-Δ+κ^2$ was one. They observed that this value was achieved on a two-parameter family, $\mathcal{O}$, of geometrically distinct ovals containing the round circle and c…