Related papers: Type-0 singularities in the network flow -- Evolut…
We consider the motion by curvature of a network of curves in the plane and we discuss existence, uniqueness, singularity formation and asymptotic behavior of the flow.
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…
We present a collection of results on the evolution by curvature of networks of planar curves. We discuss in particular the existence of a solution and the analysis of singularities.
We consider the evolution by curvature of a general embedded network with two triple junctions. We classify the possible singularities and we discuss the long time existence of the evolution.
We study the formation of singularities for the curvature flow of networks when the initial data is symmetric with respect to a pair of perpendicular axes and has two triple junctions. We show that, in this case, the set of singular times…
We prove that the curvature flow of an embedded planar network of three curves connected through a triple junction, with fixed endpoints on the boundary of a given strictly convex domain, exists smooth until the lengths of the three curves…
We prove existence and uniqueness of the motion by curvatureof networks in $\mathbb{R}^n$ when the initial datum is of class $W^{2-\frac{2}{p}}_p$, with triple junction where the unit tangent vectors to the concurring curves form angles of…
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 a network in the Euclidean plane that consists of three distinct half-lines with common start points. From that network as initial condition, there exists a network that consists of three curves that all start at one point,…
We define Type I singularities for the mean curvature flow associated to a density $\psi$ ($\psi$MCF) and describe the blow-up at singular time of these singularities. Special attention is paid to the case where the singularity come from…
We consider convex symmetric lens-shaped networks in R^2 that evolve under curve shortening flow. We show that the enclosed convex domain shrinks to a point in finite time. Furthermore, after appropriate rescaling the evolving networks…
We provide a long-time existence and sub-convergence result for the elastic flow of a three network in $\mathbb{R}^{n}$ under some mild topological assumptions. The evolution is such that the sum of the elastic energies of the three curves…
Singularities of the mean curvature flow of an embedded surface in R^3 are expected to be modelled on self-shrinkers that are compact, cylindrical, or asymptotically conical. In order to understand the flow before and after the singular…
We provide a condition for spatial curves which rules out the development of a type I singularity. The condition is that after the last time for which an inflection point develops, if the torsion is ever everywhere non-negative, the curve…
We study the evolution of the network properties of a populated network embedded in a genotype space characterised by either a low or a high number of potential links, with particular emphasis on the connectivity and clustering. Evolution…
In this paper we give sufficient conditions that guarantee the meancurvature flow with free boundary on an embedded rotationally symmetric double cone develops a Type 2 curvature singularity. We additionally prove that Type 0 singularities…
In this paper we investigate networks whose evolution is governed by the interaction of a random assembly process and an optimization process. In the first process, new nodes are added one at a time and form connections to randomly selected…
We study the curvature flow of planar nonconvex lens-shaped domains, considered as special symmetric networks with two triple junctions. We show that the evolving domain becomes convex in finite time; then it shrinks homothetically to a…
Mean curvature flow is the most natural evolution equation in extrinsic geometry, and shares many features with Hamilton's Ricci flow from intrinsic geometry. In this lecture series, I will provide an introduction to the mean curvature flow…
Many complex systems can be described in terms of networks of interacting units. Recent studies have shown that a wide class of both natural and artificial nets display a surprisingly widespread feature: the presence of highly heterogeneous…