arXiv research

New moving plane method for varifolds promotes smoothness from boundary to interior.

We review classical results where the method of the moving planes has been used to prove symmetry properties for overdetermined PDE's boundary value problems (such as Serrin's overdetermined problem) and for rigidity problems in geometric analysis (like Alexandrov soap bubble Theorem), and we give an overview of some r…

Paper introduces clock moves for plane graphs and proves Alexander polynomial properties.

Extends symmetry and rigidity to surfaces with soap film-like singularities.

Proves rotational symmetry for Serrin-type problems in doubly connected domains.

New moves for singular knots identified and described.

A plane curve is a knot diagram in which each crossing is replaced by a 4-valent vertex, and so are dual to a subset of planar quadrangulations. The aim of this paper is to introduce a new tool for sampling diagrams via sampling of plane curves. At present the most efficient method for sampling diagrams is rejection sa…

In this paper we study the general affine geometry of curves in affine space $A^2$. For a regular plane curves we define two kinds of moving frames. The first is of minimal order in all moving frames.The second is the Frenet moving frame. We get the moving equations of these moving frames. And we prove that curvature a…

In this paper, by the method of moving planes, we prove the symmetry result which says that classical solutions of Monge-Ampere system in the whole plane are symmetric about some point. Our system under consideration comes from the differential geometry problem.

We express invariants of Finsler manifolds in a geometrical way by means of using moving planes and their associated Jacobi curves, which are curves in a fixed homogeneous Grassmann manifold. Some applications are given.

A {\em blink} is a plane graph with an arbitrary bipartition of its edges. As a consequence of a recent result of Martelli, I show that the homeomorphisms classes of closed oriented 3-manifolds are in 1-1 correspondence with specific classes of blinks. In these classes, two blinks are equivalent if they are linked by a…

We consider an evolving plane curve with two endpoints that can move freely on the $x$-axis with generating constant contact angles. We discuss the asymptotic behavior of global-in-time solutions when the evolution of this plane curve is governed by area-preserving curvature flow equation. The main result shows that an…

New method studies moving points on curves using rotating frames.

A proof that hyperbolic plane cannot be immersed in Euclidean 3-space.

Cactus doodles are geometric objects derived from cactus groups.

Paper proves uniqueness of catenary cylinders based on their asymptotic shape.

In this paper, by the method of moving planes, we establish the monotonicity and symmetry properties of convex solutions for Monge-Ampere systems on bounded smooth planar domains.

In 2001, Oestlund conjectured that Reidemeister moves 1 and 3 are sufficient to describe a homotopy from any generic immersion from the circle into the plane to the standard embedding of the circle. We show that this conjecture is false.

The paper proves a nonlocal version of the Alexandrov Theorem for smooth boundaries.

The paper discusses Nirenberg's work on geometric problems and his personality.

In this note, we study Liouville type theorem for conformal Gaussian curvature equation (also called the mean field equation) $-Δu=K(x)e^u, in R^2$ where $K(x)$ is a smooth function on $R^2$. When $K(x)=K(x_1)$ is a sign-changing smooth function in the real line $R$, we have a non-existence result for the finite to…

Paper defines curvature equivalence for Legendre curves in a plane.

Study knot diagrams on a sphere without vertical lines, focusing on minimal crossings.

A geometric triangulation of a Riemannian manifold is a triangulation where the interior of each simplex is totally geodesic. Bistellar moves are local changes to the triangulation which are higher dimensional versions of the flip operation of triangulations in a plane. We show that geometric triangulations of a compac…

Study of nonlocal mean curvature and symmetry of surfaces.

In this paper, we prove the mean-convex neighborhood conjecture for neck singularities of the mean curvature flow in $\mathbb{R}^{n+1}$ for all $n\geq 3$: we show that if a mean curvature flow $\{M_t\}$ in $\mathbb{R}^{n+1}$ has an $S^{n-1}\times \mathbb{R}$ singularity at $(x_0,t_0)$, then there exists an $\varepsilon…

Explains a theorem about curves in the plane.

One-parameter hyperbolic planar motion was first studied by S. Y$\ddot{\texttt{u}}$ce and N. Kuruo$\tilde{\texttt{g}}$lu. Moreover, they analyzed the relationships between the absolute, relative and sliding velocities of one-parameter hyperbolic planar motion as well as the related pole curves, \cite{Yuc}. One-paramete…

The paper studies how the crossing number of graphs changes with a specific transformation called ΔY-move.

A new formula detects differences between counterexamples and standard embeddings of circles.

We prove that there are no networks homeomorphic to the Greek "theta" letter (a double cell) embedded in the plane with two triple junctions with angles of $120$ degrees, such that under the motion by curvature they are self-similarly shrinking. This fact completes the classification of the self-similarly shrinking net…

We prove the first nontrivial worst-case lower bounds for two closely related problems. First, $Ω(n^{3/2})$ degree-1 reductions, series-parallel reductions, and $Δ$Y transformations are required in the worst case to reduce an $n$-vertex plane graph to a single vertex or edge. The lower bound is achieved by any planar g…

Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called homotopy moves. We prove that simplifying a planar closed curve with $n$ self-crossings requires $Θ(n^{3/2})$ homotopy moves in the worst case. Our algorithm improves the best previou…

In this work, we study plane and spherical curves in Euclidean and Lorentz-Minkowski 3-spaces by employing rotation minimizing (RM) frames. By conveniently writing the curvature and torsion for a curve on a sphere, we show how to find the angle between the principal normal and an RM vector field for spherical curves. L…

In this paper, we prove than given two cubic knots $K_1$, $K_2$ in $\mathbb{R}^3$, they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister moves for classical tame knots. We use the fact that a cubic knot is determined by…

Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Re…

A mechanical linkage is a mechanism made of rigid rods linked together by flexible joints, in which some vertices are fixed and others may move. The partial configuration space of a linkage is the set of all the possible positions of a subset of the vertices. We characterize the possible partial configuration spaces of…

We prove the following quantitative version of the celebrated Soap Bubble Theorem of Alexandrov. Let $S$ be a $C^2$ closed embedded hypersurface of $\mathbb{R}^{n+1}$, $n\geq1$, and denote by $osc(H)$ the oscillation of its mean curvature. We prove that there exists a positive $\varepsilon$, depending on $n$ and upper …

The tremendous growth of positioning technologies and GPS enabled devices has produced huge volumes of tracking data during the recent years. This source of information constitutes a rich input for data analytics processes, either offline (e.g. cluster analysis, hot motion discovery) or online (e.g. short-term forecast…

A mechanical linkage is a mechanism made of rigid rods linked together by flexible joints, in which some vertices are fixed and others may move. The partial configuration space of a linkage is the set of all the possible positions of a subset of the vertices. We characterize the possible partial configuration spaces of…

Noether's Theorem yields conservation laws for a Lagrangian with a variational symmetry group. The explicit formulae for the laws are well known and the symmetry group is known to act on the linear space generated by the conservation laws. The aim of this paper is to explain the mathematical structure of both the Euler…

It is well known that surface-links in 4-space can be presented by diagrams on the plane of 4-valent spatial graphs with makers on the vertices, called marked graph diagrams. In this paper we extend the method of presenting surface-links by marked graph diagrams to presenting immersed surface-links. We also give some m…

In \cite{Mul} one-parameter planar motion was first introduced and the relations between absolute, relative, sliding velocities (and accelerations) in the Euclidean plane $\mathbb{E}^2$ were obtained. Moreover, the relations between the Complex velocities one-parameter motion in the Complex plane were provided by \cite…

This thesis is devoted to algorithmic aspects of the implementation of Cartan's moving frame method to the problem of the equivalence of submanifolds under a Lie group action. We adopt a general definition of a moving frame as an equivariant map from the space of submanifolds to the group itself and introduce two algor…

Study motion planning for points avoiding obstacles in a plane.

We provide sharp stability estimates for the Alexandrov Soap Bubble Theorem in the hyperbolic space. The closeness to a single sphere is quantified in terms of the dimension, the measure of the hypersurface and the radius of the touching ball condition. As consequence we obtain a new pinching result for hypersurfaces i…

Suppose curves are moving by curvature in a plane, but one embeds the plane in $R^3$ and looks at the plane from an angle. Then circles shrinking to a round point would appear to be ellipses shrinking to an ``elliptical point,'' and the surface energy would appear to be anisotropic as would the mobility. The result of …

The study examines special domains in S^2 supporting specific solutions to a PDE.