Related papers: Highly Degenerate Harmonic Mean Curvature Flow

We study the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain in the Euclidean plane. Under certain conditions on the initial curve the flow does not develop any singularity, and it…

We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1} (n\geq 2)$ with the speed given by arbitrary positive power $\alpha$ of the Gauss curvature. We prove that if the…

We study noncompact surfaces evolving by mean curvature flow. Without any symmetry assumptions, we prove that any solution that is $C^3$-close at some time to a standard neck will develop a neckpinch singularity in finite time, will become…

We consider the evolution by mean curvature flow of a closed submanifold of the complex projective space. We show that, if the submanifold has small codimension and satisfies a suitable pinching condition on the second fundamental form,…

In this article, we extend Huisken's theorem that convex surfaces flow to round points by mean curvature flow. We construct certain classes of mean convex and non-mean convex hypersurfaces that shrink to round points and use these…

The skew mean curvature flow(SMCF), which origins from the study of fluid dynamics, describes the evolution of a codimension two submanifold along its binormal direction. We study the basic properties of the SMCF and prove the existence of…

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 study the averaged mean curvature flow, also called the volume preserving mean curvature flow, in the particular setting of axisymmetric surfaces embedded in R^3 satisfying periodic boundary conditions. We establish analytic…

This work introduces the framed curvature flow, a generalization of both the curve shortening flow and the vortex filament equation. Here, the magnitude of the velocity vector is still determined by the curvature, but its direction is given…

In this paper, we study a class of flows of closed, star-shaped hypersurfaces in hyperbolic space $\mathbb{H}^{n+1}$ with speed $(\sinh r)^{{\alpha}/{\beta}} \sigma_{k}^{{1}/{\beta}}$, where $\sigma_{k}$ is the $k$-th elementary symmetric…

In this paper we complete the study started in [Pi2] of evolution by inverse mean curvature flow of star-shaped hypersurface in non-compact rank one symmetric spaces. We consider the evolution by inverse mean curvature flow of a closed,…

We study the evolution of a Jordan curve on the plane by curvature flow, also known as curve shortening flow, and by level-set flow, which is a weak formulation of curvature flow. We show that the evolution of the curve depends continuously…

We study a geometric flow on curves, immersed in $\mathbb{R}^3$, that have strictly positive torsion. The evolution equation is given by $$X_{t}=\frac{1}{\sqrt{\tau}} \textbf{B}$$ where $\tau$ is the torsion and $\textbf{B}$ is the unit…

We study graphical mean curvature flow of complete solutions defined on subsets of Euclidean space. We obtain smooth long time existence. The projections of the evolving graphs also solve mean curvature flow. Hence this approach allows to…

We establish a sharp rate of convergence for a free-boundary curve shortening flow in a convex domain in $\mathbb{R}^{2}$ which converges in finite time to a round half-point.

We investigate the area-preserving mean-curvature-type motion of a two-dimensional lattice crystal obtained by coupling constrained minimizing movements scheme introduced by Almgren, Taylor and Wang with a discrete-to-continuous analysis.…

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…

In this paper we study smooth solutions to a fractional mean curvature flow equation. We establish a comparison principle and consequences such as uniqueness and finite extinction time for compact solutions. We also establish evolutions…

We study a stochastically perturbed mean curvature flow for graphs in $\mathbb{R}^3$ over the two-dimensional unit-cube subject to periodic boundary conditions. In particular, we establish the existence of a weak martingale solution. The…

Consider the mean curvature flow of an (n+1)-dimensional, compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the m-th homotopy group of the complementary region can die only if…