Related papers: The length-constrained ideal curve flow
Under mean curvature flow, a closed, embedded hypersurface $M(t)$ becomes singular in finite time. For certain classes of mean-convex mean curvature flows, we show the continuity of the first singular time $T$ and the limit set "$M(T)$",…
The curve shortening flow is a geometric heat equation for curves and provides an accessible setting to illustrate many important concepts from nonlinear partial differential equations, including maximum principle estimates, monotonicity…
The behavior of the curve shortening flow has been extensively studied. Gage, Hamilton, and Grayson proved that, under the curve shortening flow, an embedded closed curve in the Euclidean plane becomes convex after a finite time and then…
We consider the inverse mean curvature flow by parallel hypersurfaces in space forms. We show that such a flow exists if and only if the initial hypersurface is isoparametric. The flow is characterized by an algebraic equation satisfied by…
We propose a weak formulation for the binormal curvature flow of curves in $\R^3.$ This formulation is sufficiently broad to consider integral currents as initial data, and sufficiently strong for the weak-strong uniqueness property to…
A comparison theorem for the isoperimetric profile on the universal cover of surfaces evolving by normalised Ricci flow is proven. For any initial metric, a model comparison is constructed that initially lies below the profile of the…
In arXiv:2205.02920 a variant of the classical elastic flow for closed curves in $\mathbb{R}^{n}$ was introduced, that is more suitable for numerical purposes. Here we investigate the long-time properties of such evolution demonstrating…
We study the length-preserving elastic flow of curves in arbitrary codimension with free boundary on hypersurfaces. This constrained gradient flow is given by a nonlocal evolution equation with nonlinear higher-order boundary conditions. We…
Hyperbolic curvature flow is a geometric evolution equation that in the plane can be viewed as the natural hyperbolic analogue of curve shortening flow. It was proposed by Gurtin and Podio-Guidugli (1991) to model certain wave phenomena in…
Ancient solutions arise in the study of parabolic blow-ups. If we can categorize ancient solutions, we can better understand blow-up limits. Based on an argument of Giga and Kohn, we give a Liouville-type theorem restricting ancient,…
In this paper, the general formulation for inextensible flows of curves on oriented surface in $\mathbb{R}^3 $ is investigated. The necessary and sufficient conditions for inextensible curve flow lying an oriented surface are expressed as a…
Principal curves are defined as parametric curves passing through the "middle" of a probability distribution in R^d. In addition to the original definition based on self-consistency, several points of view have been considered among which a…
In this paper, we study a family of curves on $S^2$ that defines a two-dimensional smooth projective plane. We use curve shortening flow to prove that any two-dimensional smooth projective plane can be smoothly deformed through a family of…
We study the evolution of curves with fixed length and clamped boundary conditions moving by the negative $L^2$-gradient flow of the elastic energy. For any initial curve lying merely in the energy space we show existence and parabolic…
Let $M$ be a closed Riemannian manifold with a parallel 1-form $\Omega$. We prove two theorems about the curve shortening flow in $M$. One is that the {\csf} $\ct$ in $M$ exists for all $t$ in $[0, \infty)$, if it satisfies $\Omega(T)\geq…
Motivated by Legendrian curve shortening flows in $\mathbb{R}^{3}$, we study the curve shortening flow of figure-eight curves in the plane. We show that, under some symmetry and curvature conditions, a figure-eight curve will shrink to a…
We collect and present in a unified way several results in recent years about the elastic flow of curves and networks, trying to draw the state of the art of the subject. In particular, we give a complete proof of global existence and…
Streets and Tian introduced pluriclosed flow and symplectic curvature flow in recent years. Here we construct a curvature flow to unify these two flows. We show the short time existence of our flow and exhibit an obstruction to long time…
We consider motion by anisotropic curvature of a network of three curves immersed in the plane meeting at a triple junction and with the other ends fixed. We show existence, uniqueness and regularity of a maximal geometric solution and we…
We consider the motion by curvature of a network of smooth curves with multiple junctions in the plane, that is, the geometric gradient flow associated to the length functional. Such a flow represents the evolution of a two--dimensional…