Related papers: Stationary sets of the mean curvature flow with a …

We consider the flat flow solutions of the mean curvature equation with a forcing term in the plane. We prove that for every constant forcing term the stationary sets are given by a finite union of disks with equal radii and disjoint…

In this paper, we consider the mean curvature flow of convex hypersurfaces in Euclidean spaces with a general forcing term. We show that the flow may shrink to a point in finite time if the forcing term is small, or exist for all times and…

This paper investigates geometric properties and well-posedness of a mean curvature flow with volume-dependent forcing. With the class of forcing which bounds the volume of the evolving set away from zero and infinity, we show that a strong…

We consider the forced mean curvature flow in 2-d, finite range of dependence and positive random forcing. We prove flatness and existence of effective speed for initially flat propagating fronts. This is the analogue, in random media, of a…

We consider the long-time behavior of the mean curvature flow in heterogeneous media with periodic fibrations, modeled as an additive driving force. Under appropriate assumptions on the forcing term, we show existence of generalized…

We prove uniqueness of tangent cones for forced mean curvature flow, at both closed self-shrinkers and round cylindrical self-shrinkers, in any codimension. The corresponding results for mean curvature flow in Euclidean space were proven by…

In this paper we establish a new stability result for the smooth volume preserving mean curvature flow in flat torus $\mathbb T^n$ in low dimensions $n=3,4$. The result says roughly that if the initial set is near to a strictly stable set…

We consider the volume constrained fractional mean curvature flow of a nearly spherical set, and prove long time existence and asymptotic convergence to a ball. The result applies in particular to convex initial data, under the assumption…

We show that under suitable non-degeneracy conditions, complete gradient flow lines of the scalar curvature functional of a riemannian manifold perturb into eternal forced mean curvature flows with large forcing term.

The recent work of Morini-Oronzio-Spadaro and the third author shows that, in three dimensions, a flat-flow solution of the volume-preserving mean curvature flow that converges to a single ball, which is the case for instance when the…

An existence and uniqueness result, up to fattening, for crystalline mean curvature flows with forcing and arbitrary (convex) mobilities, is proven. This is achieved by introducing a new notion of solution to the corresponding level set…

We complete the theoretical framework required for the construction of a Morse homology theory for certain types of forced mean curvature flows. The main result of this paper describes the asymptotic behaviour of these flows as the forcing…

We study the motion of a droplet evolving by mean curvature with volume constraint and contact angle condition on a half space. We prove the existence of a global-in-time weak solution, called the flat flow. A difficulty arises when we…

In this paper, we prove short time existence and uniqueness of smooth evolution by mean curvature in $\mathbb{R}^{n+1}$ starting from any $n$-dimensional $(\varepsilon,R)$-Reifenberg flat set with $\varepsilon$ sufficiently small. More…

The aim of this paper is to introduce a notion of mean curvature flow soliton general enough to encompass target spaces of constant sectional curvature, Riemannian products or, in increasing generality, warped product spaces.

We prove that, in the flat torus and in any dimension, the volume-preserving mean curvature flow and the surface diffusion flow, starting $C^{1,1}-$close to a strictly stable critical set of the perimeter $E$, exist for all times and…

We study the mean curvature flow with given non-smooth transport term and forcing term, in suitable Sobolev spaces. We prove the global existence of the weak solutions for the mean curvature flow with the terms, by using the modified…

We flow a hypersurface in Euclidean space by mean curvature flow with a Neumann boundary condition, where the boundary manifold is any torus of revolution. If we impose the conditions that the initial manifold is compatible and does not…

The symmetry group of the mean curvature flow in general ambient Riemannian manifolds is determined, based on which we define generalized solitons to the mean curvature flow. We also provide examples of homothetic solitons in non-Euclidean…

We study the global existence and stability of surface diffusion flow (the normal velocity is given by the Laplacian of the mean curvature) of smooth boundaries of subsets of the $n$--dimensional flat torus. More precisely, we show that if…