arXiv research

Classifies 85 tie knots into mathematical categories.

Paper proposes a smart neck-band for detecting neck postures using integrated kinematic and kinetic data.

Study on harmonic maps from surfaces with energy bounds and neck domains.

The paper extends energy identities and neck existence for ε-harmonic maps.

Study neck pinches in Lagrangian flows, proving stability and introducing new singularities.

Paper solves long neck problem on odd-dimensional spin manifolds.

Paper proves energy identity and no-neck property for special harmonic maps.

A multi-neck spacetime wormhole is constructed with a simple metric tensor.

Derives generalizations of the long neck principle and spectral width inequality.

In this paper we study motion of surfaces of revolution under the mean curvature flow. For an open set of initial conditions close to cylindrical surfaces we show that the solution forms a "neck" which pinches in a finite time at a single point. We also obtain a detailed description of the neck pinching process.

In this paper, we prove some refined estimate in the neck region when a sequence of harmonic maps from surfaces blow up. The new estimate allows us to see the shape of the center of the neck region. As an application, we prove an inequality about the nullity and index when blow-up occurs.

Study geodesics on neck-degenerate manifolds, focusing and winding behavior observed.

The shape equation and linking conditions for a vesicle with two-phase domains are derived. We refine the conjecture on the general neck condition for the limit shape of a budding vesicle proposed by Jülicher and Lipowsky [Phys. Rev. Lett. \textbf{70}, 2964 (1993); Phys. Rev. E \textbf{53}, 2670 (1996)], and then we us…

We prove the energy identity and the no neck property for a sequence of smooth extrinsic polyharmonic maps with bounded total energy.

We find calibrated submanifolds in neck manifolds. Particularly, we obtain a calibrated submanifold in the Lagrangian self-expander constructed by Joyce, Lee and Tsui.

Lipid necks, large curvature bridges, are shown to be metastable.

In this paper we are dealing with mean curvature flow with surgeries of two-convex hypersurfaces. The main focus is to expand on the discussion in Section $3$ of Mean Curvature Flow with Surgeries of Two-Convex Hypersurfaces by Huisken and Sinestrari. Firstly we wish to establish how the neck detection lemma allows us …

We will give a weak energy identity for Sacks-Uhlenbeck approximation of harmonic maps and calculate the length of the necks.

Study controls curvature in Ricci flows using necks.

In this paper we introduce the tied links, i.e. ordinary links provided with some ties between strands. The motivation for introducing such objects originates from a diagrammatical interpretation of the defining generators of the so-called algebra of braids and ties; indeed, one half of such generators can be interpret…

Tied links and the tied braid monoid were introduced recently by the authors and used to define new invariants for classical links. Here, we give a version purely algebraic-combinatoric of tied links. With this new version we prove that the tied braid monoid has a decomposition like a semi--direct group product. By usi…

Study tied links in various 3-manifolds, introducing new groups and proving theorems.

Mean curvature flow shows singularities on smooth surfaces.

The paper proves the existence of maxfaces with multiple swallowtails and planar ends.

Proves principles and estimates for initial data sets in Einstein equations.

We study moduli spaces of Seiberg-Witten monopoles over spin^c Riemannian 4-manifolds with long necks and/or tubular ends. This first part discusses compactness, exponential decay, and transversality. As applications we prove two vanishing theorems for Seiberg-Witten invariants.

New invariant for tied links connects states without resolution dependence.

New monoids tied to symmetric group and Jones/Brauer monoids discovered.

Constructs minimal immersions with singularities.

Paper derives explicit formulas for AJ-bracket of tied links.

Study of pseudo knots, links, and knotoids with braiding and L-moves.

We establish a gluing theorem for monopoles over 4--manifolds containing long necks. The theorem is stated in terms of an ungluing map defined explicitly in terms of data that appear naturally in applications. Orientations of moduli spaces are handled using Benevieri--Furi's concept of orientations of Fredholm operator…

We examine a Type-1 neck pinch singularity in simplicial Ricci flow (SRF) for an axisymmetric piecewise flat 3-dimensional geometry with 3-sphere topology. SRF was recently introduced as an unstructured mesh formulation of Hamilton's Ricci flow (RF). It describes the RF of a piecewise-flat simplicial geometry. In this …

3D Ricci flows have bounded diameter before Type I singularities.

It is a fundamental open problem for the mean curvature flow, and in fact for many partial differential equations, whether or not all blowup limits are selfsimilar. In this short note, we prove that for the mean curvature flow of mean convex surfaces all limit flows are selfsimilar (static, shrinking or translating) if…

We introduce the concept of tied links in the solid torus, which generalize naturally the concept of tied links in $S^3$ previously introduced by Aicardi and Juyumaya. We also define an invariant of these tied links by using skein relations, and subsequently we recover this invariant by using Jones' method over the bt-…

Researchers create minimal surfaces with Scherk ends and find catenoid limits.

Constructing translating solitons from Lagrangian Grim Reapers.

Study verifies Joyce's conjectures for circle-invariant Lagrangian surfaces.

The paper constructs harmonic 1-forms on 3-manifolds with cylindrical necks.

New proof classifies ancient flows in 3D space.

Let $M$ be a closed symplectic manifold of dimension $2n$ with non-ellipticity. We can define an almost Kähler structure on $M$ by using the given symplectic form. Hence, we have a $\G=π_1(M)$-invariant almost Kähler structure on the universal covering, $\ti M$, of $M$. Using Darboux coordinate charts, we globally defo…

Paper uses 3-circle theorem to study Willmore surfaces and prove decay estimates.

Constructs surfaces with specific topologies and curvatures.

Real Lagrangian tori in $S^2 imes S^2$ are Hamiltonian isotopic to the Clifford torus.

In earlier work, carrying out numerical simulations of the Ricci flow of families of rotationally symmetric geometries on $S3$, we have found strong support for the contention that (at least in the rotationally symmetric case) the Ricci flow for a ``critical'' initial geometry - one which is at the transition point bet…

We define two new invariants for tied links. One of them can be thought as an extension of the Kauffman polynomial and the other one as an extension of the Jones polynomial which is constructed via a bracket polynomial for tied links. These invariants are more powerful than both the Kauffman and the bracket polynomials…

Classifies ancient noncollapsed flows in 4D space.