Related papers: Evolving convex curves by a generalized length-pre…

We consider graphical solutions to mean curvature flow and obtain a stability result for homothetically expanding solutions coming out of cones of positive mean curvature: If another solution is initially close to the cone at infinity, then…

In this paper, we prove the short-time existence of hyperbolic inverse (mean) curvature flow (with or without the specified forcing term) under the assumption that the initial compact smooth hypersurface of $\mathbb{R}^{n+1}$…

We study the group of automorphisms of the affine plane preserving some given curve, over any field. The group is proven to be algebraic, except in the case where the curve is a bunch of parallel lines. Moreover, a classification of the…

A new simple Lagrangian method with favorable stability and efficiency properties for computing general plane curve evolutions is presented. The method is based on the flowing finite volume discretization of the intrinsic partial…

This paper concerns the inverse mean curvature flow of convex hypersurfaces which are Lipschitz in general. After defining a weak solution, we study the evolution of the singularity by looking at the blow-up tangent cone around each…

We study families of smooth immersed regular planar curves $ \alpha : \left [-1,1 \right ]\times \left [0,T \right )\to \mathbb{R}^{2}$ satisfying the fourth order nonlinear curve diffusion flow with generalised Neumann boundary conditions…

We describe the curves of constant (geodesic) curvature and torsion in the three-dimensional round sphere. These curves are the trajectory of a point whose motion is the superposition of two circular motions in orthogonal planes. The global…

We consider a diffused interface version of the volume-preserving mean curvature flow in the Euclidean space, and prove, in every dimension and under natural assumptions on the initial datum, exponential convergence towards single "diffused…

We consider the evolution of fronts by mean curvature in the presence of obstacles. We construct a weak solution to the flow by means of a variational method, corresponding to an implicit time-discretization scheme. Assuming the regularity…

The equation of a motion of curves in the projective plane is deduced. Local flows are defined in terms of polynomial differential functions. A family of local flows inducing the Kaup-Kupershmidt hierarchy is constructed. The integration of…

In this paper, we generalize Chow-Luo's combinatorial Ricci flow to inversive distance circle packing setting. Although the solution to the generalized flow may develop singularities in finite time, we can always extend the solution so as…

This survey reviews some facts about nonnegativity conditions on the curvature tensor of a Riemannian manifold which are preserved by the action of the Ricci flow. The text focuses on two main points. First we describe the known examples of…

We prove that the static convexity is preserved along two kinds of locally constrained curvature flows in hyperbolic space. Using the static convexity of the flow hypersurfaces, we prove new family of geometric inequalities for such…

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.

It has long been conjectured that starting at a generic smooth closed embedded surface in R^3, the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or…

We show existence of ancient solutions to the rescaled mean curvature flow starting from a given asymptotically conical self-expander. These are examples of mean curvature flows coming out of cones that are not self-similar. We also show a…

We show that the mean curvature flow for a closed and rotationally symmetric surface can be formulated as an evolution problem consisting of an evolution equation for the square of the function whose graph is rotated and two ODEs describing…

In this paper we analyse the Euler implicit scheme for the volume preserving mean curvature flow. We prove the exponential convergence of the scheme to a finite union of disjoint balls with equal volume for any bounded initial set with…

We study the lengths of curves passing through a fixed number of points on the boundary of a convex shape in the plane. We show that for any convex shape $K$, there exist four points on the boundary of $K$ such that the length of any curve…

In this paper, we study the evolution of smooth, closed planar curves under a fourth order biharmonic flow with an external forcing term. Such flows arise naturally in the theory of biharmonic maps and geometric variational problems…