arXiv research

Flow turns star-shaped curves into circles.

Paper proves conjecture about star-shaped curves evolving under GAPF, but not always preserves star shape.

Curves become nearly circular over time without initial assumptions.

Study curve shortening flow on Riemann surfaces with conic singularities.

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

We study the area preserving Willmore flow in an asymptotic region of an asymptotically flat manifold which is $C^{3}-$close to Schwarzschild. It was shown by Lamm, Metzger and Schulze that such an end is foliated by spheres of Willmore type. In this paper, we prove that the leaves of this foliation are stable under sm…

New curve flow preserves area and converges to a circle.

The paper studies area-preserving and length-preserving inverse curvature flow for planar curves with singularities.

Study proves existence and properties of shrinkers in area-preserving curve-shortening flow.

Study a flow preserving area of plane curves, ending in a circle.

By the curve shortening flow, the only closed embedded contracting self-similar solutions are circles: we give a very short and intuitive geometric proof of this basic and classical result using an idea of Gage.

In this paper, we consider a kind of area preserving non-local flow for convex curves in the plane. We show that the flow exists globally, the length of evolving curve is non-increasing, and the curve converges to a circle in C^{\infty} sense as time goes into infinity.

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.\$

The blow-up rates of derivatives of the curvature function will be presented when the closed curves contract to a point in finite time under the general curve shortening flow. In particular, this generalizes a theorem of M.E. Gage and R.S. Hamilton about mean curvature flow in $\mathbb{R}^{2}$.

The paper proves the stability of a flow in Schwarzschild space.

We show that the surface area preserving mean curvature flow in Euclidean space exists for all time and converges exponentially to a round sphere, if initially the L^2-norm of the traceless second fundamental form is small (but the initial hypersurface is not necessarily convex).

Study shows how flat flow solutions in 2D converge to disks.

New proof of Wulff-Gage inequality with applications.

In this paper, we consider the area-preserving mean curvature flow with free Neumann boundaries. We show that for a rotationally symmetric $n$-dimensional hypersurface in $\R^{n+1}$ between two parallel hyperplanes will converge to a cylinder with the same area under this flow. We use the geometric properties and the m…

The paper examines flows that preserve area and length in hyperbolic geometry.

The paper studies the stability of volume and area preserving mean curvature flows in Schwarzschild and asymptotic Schwarzschild spaces.

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

Let f:Σ_1 --> Σ_2 be an area preserving diffeomorphism between compact Riemann surfaces of constant curvature. The graph of f can be viewed as a Lagrangian submanifold in Σ_1\times Σ_2. This article discusses a canonical way to deform f along area preserving diffeomorphisms. This deformation process is realized through…

We consider the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain or at a straight line. We give a criterion on initial curves that guarantees the appearance of a singularity in finite time. We prove that the singularity is of type II. Furthermore, if these initial c…

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…

Curve shortening in metric-affine plane shrinks convex curves to points.

Small bubbles sliding on a boundary maintain half-spherical shape.

Paper finds new criteria for conjugate points in fluid flows.

Curve shortening flow shrinks curves to points under certain conditions.

This article gives an alternative approach to the self-shrinking and self-expanding solutions of the curve shortening flow, which are related to singularity formation of the mean curvature flow. The motivation for the self-similar solutions arises from natural area preserving rescaling. Further we describe the self-sim…

The paper studies mean curvature flow of spacelike-convex submanifolds in pseudo-Euclidean space.

Study on curve shortening flow in 3D space curves, showing convexity preservation and avoidance principle.

Area-preserving diffeomorphisms of a 2-disc can be regarded as time-1 maps of (non-autonomous) Hamiltonian flows on solid tori, periodic flow-lines of which define braid (conjugacy) classes, up to full twists. We examine the dynamics relative to such braid classes and define a braid Floer homology. This refinement of t…

Study shows how curved surfaces evolve smoothly to spherical shapes.

The paper studies curve shortening flows on non-convex surfaces.

We provide explicit examples which show that mean convexity (i.e. positivity of the mean curvature) and positivity of the scalar curvature are non-preserved curvature conditions for hypersurfaces of the Euclidean space evolving under either the volume- or the area preserving mean curvature flow. The relevance of our ex…

We consider a convex Euclidean hypersurface that evolves by a volume or area preserving flow with speed given by a general nonhomogeneous function of the mean curvature. For a broad class of possible speed functions, we show that any closed convex hypersurface converges to a round sphere. The proof is based on the mono…

Classifies central extensions for area-preserving diffeomorphisms and shows they are fuzzy sphere limits.

General area-preserving motion of polygonal curves is formulated as a system of ODEs. Solution polygonal curves belong to a prescribed polygonal class, which is similar to the admissible class used in the crystalline curvature flow. The ODEs are discretized implicitly in time keeping a given constant area speed while s…

Constructs foliations of lightcones using surfaces of constant spacetime mean curvature.

Classifies nets with area-preserving transformations into two types.

The pure braid group cannot be realized as area-preserving homeomorphisms.

Study on stability of surfaces in null cones under area-preserving variations.

Study vortex loops as coadjoint orbits of diffeomorphisms.

Short proof shows infinite diameter for surface diffeomorphisms.

We study polygonal analogues of several moving boundary problems and their time discretization which preserves the constant area speed property. We establish various polygonal analogues of geometric formulas for moving boundaries and make use of the geometric formulas for our numerical scheme and its analysis of genera…

New results on geometry of area-preserving diffeomorphisms using braids.

BiLipschitz mappings can be extended to preserve area.