Related papers: Monotonicity of the modulus under curve shortening…

In this paper we consider the steepest descent $H^{-1}$-gradient flow of the length functional for immersed plane curves, known as the curve diffusion flow. It is known that under this flow there exist both initially immersed curves which…

In this paper we consider the anisotropic curve shortening flow in the plane in the presence of an ambient force. We consider force fields in which all their derivatives are bounded in the $L^{\infty}$ sense. We prove that closed embedded…

In this paper we study the curvature flow of a curve in a plane endowed with a minkowskian norm whose unit ball is smooth. We show that many of the properties known in the euclidean case can be extended (with due adaptations) to this new…

In this paper, we study curve shortening flows on rotational surfaces in $\mathbb{R}^3$. We assume that the surfaces have negative Gauss curvatures and that some condition related to the Gauss curvature and the curvature of embedded curve…

We introduce a flow in the space of constant width bodies in three-dimensional Euclidean space that simultaneously increases the volume and decreases the circumradius of the shape as time increases. Starting from any initial constant width…

In this paper we introduce two $1/\kappa^{n}$-type ($n\ge1$) curvature flows for closed convex planar curves. Along the flows the length of the curve is decreasing while the enclosed area is increasing. And finally, the evolving curves…

Recently Andrews and Bryan [3] discovered a comparison function which allows them to shorten the classical proof of the well-known fact that the curve shortening flow shrinks embedded closed curves in the plane to a round point. Using this…

In this paper, we study a new area-preserving curvature flow for closed convex planar curves. This flow will decrease the length of the evolving curve and make the curve more and more circular during the evolution process. And finally, the…

We prove that a simple closed plane curve with curvature at most 1 and diameter at least 4, surrounds two disjoint open unit disks. We also pose an open question relating the diameter lower bound to the length of the curve.

We study the spectral stability of Dirichlet eigenvalues on an embedded annulus whose boundary evolves by curve shortening flow while the ambient surface evolves under the two dimensional Ricci flow using variational formulas, Rellich--type…

We consider embedded, smooth curves in the plane which are either closed or asymptotic to two lines. We study their behaviour under curve shortening flow with a global forcing term. Firstly, we prove an analogue to Huisken's distance…

In this paper, we consider a new length preserving curve flow for convex curves in the plane. We show that the global flow exists, the area of the region bounded by the evolving curve is increasing, and the evolving curve converges to the…

We consider a motion of non-closed planar curves with infinite length. The motion is governed by a steepest descent flow for the geometric functional which consists of the sum of the length functional and the total squared curvature. We…

We present a second order accurate in time numerical scheme for curve shortening flow in the plane that is unconditionally monotone. It is a variant of threshold dynamics, a class of algorithms in the spirit of the level set method that…

A nonlocal curvature flow is introduced to evolve locally convex curves in the plane. It is proved that this flow with any initial locally convex curve has a global solution, keeping the local convexity and the elastic energy of the…

We find a new monotone increasing quantity along smooth solutions to the inverse mean curvature flow in $\mathbb R^n$. As an application, we derive a sharp geometric inequality for mean convex, star-shaped hypersurfaces which relates the…

Bounds of total curvature and entropy are two common conditions placed on mean curvature flows. We show that these two hypotheses are equivalent for the class of ancient complete embedded smooth planar curve shortening flows, which are…

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…

We show that under Space Curve Shortening flow any closed immersed curve in $\mathbb R^n$ whose projection onto $\mathbb{R}^2\times\{\vec{0}\}$ is convex remains smooth until it shrinks to a point. Throughout its evolution, the projection…

We show that a curve is a soliton solution to the curve shortening flow if and only if its geodesic curvature can be written as the inner product between its tangent vector field and a fixed vector of the 3-dimensional Minkowski space. We…