arXiv research

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

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

Curves become nearly circular over time without initial assumptions.

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

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

New curve flow preserves area and converges to a circle.

Study shows curves converge to traveling waves under specific conditions.

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…

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.

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

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

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…

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

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…

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

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…

Study shows how curved surfaces evolve smoothly to spherical shapes.

Flow turns star-shaped curves into circles.

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.\$

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

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…

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…

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

Study curvature flows on pinched Hadamard surfaces, proving convexity preservation and convergence.

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

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…

We study a volume/area preserving curvature flow of hypersurfaces that are convex by horospheres in the hyperbolic space, with velocity given by a generic positive, increasing function of the mean curvature, not necessarly homogeneous. For this class of speeds we prove the exponential convergence to a geodesic sphere. …

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…

The article recovers the Smale conjecture on a Sasakian 3-sphere using Legendrian mean curvature flow.

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…

Paper finds new criteria for conjugate points in fluid flows.

We apply the Cartan equivalence method to the study of real analytic second order ODEs under the local real analytic diffeomorphism of $\C^2$ which are area-preserving. This enables us to give a characterization of the second order ODEs which are equivalent to $y^{\prime\prime}=0$ under such transformations. Moreover w…

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…

We prove that any weakly acausal curve $Γ$ in the boundary of Anti-de Sitter (2+1)-space is the asymptotic boundary of two spacelike $K$-surfaces, one of which is past-convex and the other future-convex, for every $K\in(-\infty,-1)$. The curve $Γ$ is the graph of a quasisymmetric homeomorphism of the circle if and only…

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

Classifies nets with area-preserving transformations into two types.

Let N be a complete, simply-connected surface of constant curvature κ\leq 0. Moreover, suppose that Ωand \tildeΩ are strictly convex domains in N with the same area. We show that there exists an area-preserving diffeomorphism from Ωto \tildeΩ whose graph is a minimal submanifold of N \times N.

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

In this note we obtain the characterization for asymptotic directions on various subgroups of the diffeomorphism group. We give a simple proof of non-existence of such directions for area-preserving diffeomorphisms of closed surfaces of non-zero curvature. Finally, we exhibit the common origin of the Monge-Ampere equat…

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.

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

BiLipschitz mappings can be extended to preserve area.

An area-preserving diffeomorphism of an annulus has an "action function" which measures how the diffeomorphism distorts curves. The average value of the action function over the annulus is known as the Calabi invariant of the diffeomorphism, while the average value of the action function over a periodic orbit of the di…

Survey on strong closing lemmas in Hamiltonian dynamics.