Related papers: Mean curvature flow with positive random forcing i…

The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of the mean curvature flow in this paper: a Lipschitz submanifold with small local Lipschitz norm…

We study a notion of relative entropy motivated by self-expanders of mean curvature flow. In particular, we obtain the existence of this quantity for arbitrary hypersurfaces trapped between two disjoint self-expanders asymptotic to the same…

In this note we prove the backwards uniqueness of the mean curvature flow for (codimension one) hypersurfaces in a Euclidean space. More precisely, let $F_t, \widetilde{F}_t:M^n \rightarrow \mathbb{R}^{n+1}$ be two complete solutions of the…

In this paper, we investigate two hyperbolic flows obtained by adding forcing terms in direction of the position vector to the hyperbolic mean curvature flows in \cite{klw,hdl}. For the first hyperbolic flow, as in \cite{klw}, by using…

Here, we study a level-set forced mean curvature flow with the homogeneous Neumann boundary condition. We first show that the solution is Lipschitz in time and locally Lipschitz in space. Then, under an additional condition on the forcing…

In this paper, we introduce a class of new logarithmic curvature flow. The flows are designed to embrace the monotonicity of the related functional, and the convergence of this flow would tackle the solvability of the weighted…

We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in $\R^{2n}$, we show that the parabolic…

An existence and uniqueness result, up to fattening, for a class of crystalline mean curvature flows with natural mobility is proved. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. The…

G. Pipoli and C. Sinestrari considered the mean curvature flow starting from a closed submanifold in the complex projective space. They proved that if the submanifold is of small codimension and satisfies a suitable pinching condition for…

We consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold $M\times\mathbb{R}$, where $M$ is asymptotically flat. If the initial hypersurface $F_0\subset M\times\mathbb{R}$…

We show that the discrete approximate volume preserving mean curvature flow in the flat torus $\mathbb{T}^N$ starting near a strictly stable critical set $E$ of the perimeter converges in the long time to a translate of $E$ exponentially…

We construct a mean curvature flow with surgery for submanifolds of arbitrary codimension. The theory applies to closed submanifolds satisfying a natural quadratic pinching condition, which serves as the high-codimension analogue of…

This article studies the mean curvature flow of Lagrangian submanifolds. In particular, we prove the following global existence and convergence theorem: if the potential function of a Lagrangian graph in T^{2n} is convex, then the flow…

We prove that for the mean curvature flow of two-convex hypersurfaces the intrinsic diameter stays uniformly controlled as one approaches the first singular time. We also derive sharp $L^{n-1}$-estimates for the regularity scale of the…

In [5], S\'aez and Schn\"urer studied the graphical mean curvature flow of complete hypersurfaces defined on subsets of Euclidean space. They obtained long time existence. Moreover, they provided a new interpretation of weak mean curvature…

In this paper, we introduce and study the conformal mean curvature flow of submanifolds of higher codimension in the Euclidean space $\bbr^n$. This kind of flow is a special case of a general modified mean curvature flow which is of various…

We prove new pinching estimate for the inverse curvature flow of strictly convex hypersurfaces in the space form $N$ of constant sectional curvature $K_N$ with speed given by $F^{-\alpha}$, where $\alpha\in (0,1]$ for $K_N=0,-1$ and…

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…

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 the mean curvature flow in 3-dimensional null hypersurfaces. In a spacetime a hypersurface is called null, if its induced metric is degenerate. The speed of the mean curvature flow of spacelike surfaces in a null hypersurface is…