arXiv research

In this paper we introduce the Constant Width Measure Set, which measures the constant width property of an oval, i.e. the planar simple closed strictly convex curve. We study its geometrical properties. We find the exact relation between the length and the area of the region bounded by an oval $M$. Namely, the followi…

New set class preserves Fourier series terms for planar ovals, leading to isoperimetric inequalities.

The method for approximation of planar curve by circular arcs with length preservation, proposed by I.Kh. Sabitov and A.V. Slovesnov, is analyzed. We extend the applicability of the method, and consider some corollaries, not related to the approximation problem. Inequalities for the length of a convex spiral arc with p…

We study the periods mapping from the moduli space of real hyperelliptic curves with marked point on an oriented oval to the euclidean space. The mapping arises in the analysis of Chebyshev construction used in the constrained optimization of the uniform norm of polynomials and rational functions. The decomposition of …

Classifies ancient ovals in higher dimensional mean curvature flow.

The paper classifies ovals in 4D space for a specific flow.

A surface S in R^3 has the central plane oval property (cpo) if (i) S meets at least one affine plane transversally along a strictly convex oval, and (ii) Every such transverse oval on S has central symmetry. We show that a complete, connected C^2 surface with cpo must be either a generalized cylinder, or quadric. Appl…

The paper confirms conjectures about ancient ovals and provides counterexamples.

Enhanced estimates for ancient ovals and translators in 3D and 4D.

The paper classifies ancient ovals in higher dimensions and proves their symmetry and uniqueness.

Ancient ovals are key blowup limits in 3D Ricci flow near singularities.

Classifies ancient noncollapsed flows in 4D space.

Discussing rigidity properties of conics, inspired by billiards in ellipses.

Proof of Graustein's theorem in different geometries.

The Hessian Topology is a subject having interesting relations with several areas, for instance, differential geometry, implicit differential equations, analysis and singularity theory. In this article we study the problem of realization of a real plane curve as the Hessian curve of a smooth function. The plane curves …

Study of red blood cells using elastic surface theory.

In this note we study the distribution of real inflection points among the ovals of a real non-singular hyperbolic curve of even degree. Using Hilbert's method we show that for any integers $d$ and $r$ such that $4\leq r \leq 2d^2-2d$, there is a non-singular hyperbolic curve of degree $2d$ in $\mathbb R^2$ with exactl…

The classical isoperimetric inequality in the Euclidean plane $\mathbb{R}^2$ states that for a simple closed curve $M$ of the length $L_{M}$, enclosing a region of the area $A_{M}$, one gets \begin{align*} L_{M}^2\geqslant 4πA_{M}. \end{align*} In this paper we present the improved isoperimetric inequality, which state…

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…

Two ancient solutions to Gauss curvature flow are identified for cylinders.

The paper shows that oval caustics have at least 4 cusps.

Unique asymptotics found for special geometric flows.

This paper focuses on curves and surfaces of constant width, with some additional results about general ovals. We emphasize the use of Fourier series to derive properties, some of which are known. Amongst other results, we show that the perimeter of an oval is $π$ times its average width, and provide a bound for the ra…

Macbeath gave a formula for the number of fixed points for each non-identity element of a cyclic group of automorphisms of a compact Riemann surface in terms of the universal covering transformation group of the cyclic group. We observe that this formula generalizes to determine the fixed-point set of each non-identity…

We show that the only convex ancient solutions to curve shortening flow are the stationary lines, shrinking circles, Grim Reapers and Angenent ovals, completing the classification initiated by Daskalopoulos, Hamilton and Sesum and X.-J. Wang

New inequality for odd-degree flexible curves using surface doubling.

We prove a linear in $\degω$ upper bound on the number of real zeros of the Abelian integral $I(t)=\int_{δ(t)}ω$, where $δ(t)\subset\R^2$ is the real oval $x^2y(1-x-y)=t$ and $ω$ is a one-form with polynomial coefficients.

New geometric invariant from disc intersections captures all coloured Jones polynomials.

The classical Tait-Kneser theorem states that the osculating circles of a smooth plane curve, free from curvature extrema, are pairwise disjoint. We prove a number of analogs of this theorem, e.g., for ovals of osculating cubics, osculating polynomials and trigonometric polynomials; in each case, we will obtain a non-d…

New 1-parameter family of ovals identified in 4d Ricci flow classification.

Study ancient flows in 4D, classifying based on bubble-sheet eigenvalues.

Characterizes minor-minimal separating projective planar graphs and their generalizations.

We construct an isotopy of a planar compactum that is not the restriction of an isotopy of any planar continuum.

We consider an embedded convex ancient solution $Γ_t$ to the curve shortening flow in $\mathbb{R}^2$. We prove that there are only two possibilities: the family $Γ_t$ is either the family of contracting circles, which is a type I ancient solution, or the family of evolving Angenent ovals, which correspond to a type II …

Study examines how changing regions affects planar graphs.

Paper classifies singularity models for 3D hypersurfaces in R^4.

We construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994. As time $t \rightarrow 0^-$ the solutions collapse to a round point where $0$ is the singular time. But as $t\rightarrow-\infty$ the solutions become more and more oval. Near the center the appropriately-resc…

In this paper, we introduce the notions of an iterated planar Lefschetz fibration and an iterated planar open book decomposition and prove the Weinstein conjecture for contact manifolds supporting an open book that has iterated planar pages. For $n\geq 1$, we show that a $(2n+1)$-dimensional contact manifold $M$ suppor…

New inequalities for planar convex domains' Laplacian eigenvalues.

Quasi-transitive graphs quasi-isometric to planar graphs can be upgraded to Cayley graphs.

In this short article, we prove the existence of ancient solutions of the mean curvature flow that for t -> 0 collapse to a round point, but for t -> -infinity become more and more oval: near the center they have asymptotic shrinkers modeled on round cylinders S^j x R^n-j and near the tips they have asymptotic translat…

Study of higher-dimensional contact manifolds and their properties.

Proves planar graphs' configuration spaces have highest topological complexity.

We find an invariant characterization of planar webs of maximum rank. For 4-webs, we prove that a planar 4-web is of maximum rank three if and only if it is linearizable and its curvature vanishes. This result leads to the direct web-theoretical proof of the Poincaré's theorem: a planar 4-web of maximum rank is lineari…

We characterize those planar Peano continua that are homotopy equivalent to 1-dimensional sets. While many planar Peano continua are not homotopically 1-dimensional, we prove that each has fundamental group that embeds in the fundamental group of a 1-dimensional planar Peano continuum. We leave open the following quest…

Planar multilinks prove rational singularities in surface geometry.

Paper introduces a new invariant for planar knotoids.

A graph is apex if it can be made planar by deleting a vertex, that is, $\exists v$ such that $G-v$ is planar. We define the related notions of edge apex, $\exists e$ such that $G-e$ is planar, and contraction apex, $\exists e$ such that $G/e$ is planar, as well as the analogues with a universal quantifier: $\forall v$…