相关论文: On Mean curvature flow of Singular Riemannian foli…

In this paper, we investigate the mean curvature flow of submanifolds of arbitrary codimension in $\mathbb{C}\mathbb{P}^m$. We prove that if the initial submanifold satisfies a pinching condition, then the mean curvature flow converges to a…

Michor and Mumford showed that the mean curvature flow is a gradient flow on a Riemannian structure with a degenerate geodesic distance. It is also known to destroy the uniform density of gridpoints on the evolving surfaces. We introduce a…

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…

In this paper, we show that if the mean curvature of a closed smooth embedded mean curvature flow in R^3 is of type-I, then the rescaled flow at the first finite singular time converges smoothly to a self-shrinker flow with multiplicity…

We show that a mean curvature flow starting from a compact, smoothly embedded hypersurface M remains unique past singularities, provided the singularities are of mean convex type, i.e., if around each singular point, the surface moves in…

We investigate the integral conditions to extend the mean curvature flow in a Riemannian manifold. We prove that the mean curvature flow solution with finite total mean curvature on a finite time interval $[0,T)$ can be extended over time…

We show that for generic smooth compact initial surfaces the mean curvature flow in $\mathbb{R}^3$ has spherical or nondegenerate neck pinch singularities at the first singular time. In particular the singularities at the first singular…

In this paper we consider the evolution of sets by a fractional mean curvature flow. Our main result states that for any dimension $n > 2$, there exists an embedded surface in $\mathbb R^n$ evolving by fractional mean curvature flow, which…

For every closed set $K \subset \mathbb{R}^n$ and every $m \geq 2$, we construct a mean-convex ancient solution to mean curvature flow of hypersurfaces in $\mathbb{R}^{m+n}$, with respect to a smooth Riemannian metric arbitrarily…

We show that any transversally complete Riemannian foliation F of dimension one on any possibly non-compact manifold M is tense; namely, (M,F) admits a Riemannian metric such that the mean curvature form of F is basic. This is a partial…

Let $(M, g)$ be an asymptotically flat Riemannian $3$-manifold with non-negative scalar curvature and positive mass. We show that each leaf of the canonical foliation through stable constant mean curvature surfaces of the end of $(M, g)$ is…

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…

Generalizing results of Chou and Wang \cite{1} we study the flows of the leaves $(M_{\Theta})_{\Theta>0}$ of a foliation of $\mathbb{R}^{n+1}\setminus \{0\}$ consisting of uniformly convex hypersurfaces in the direction of their outer…

We first give a general introduction to the mean curvature flow, and then discuss fundamental results established over the last 10 years that yield a precise theory for the flow through singularities in $\mathbb{R}^3$. With the aim of…

In this paper we study the uniqueness of graphical mean curvature flow. We consider as initial conditions graphs of locally Lipschitz functions and prove that in the one dimensional case solutions are unique without any further assumptions.…

In this paper, we provide the lower bounds of the first non-zero basic eigenvalue on a closed singular Riemannian manifold $(M,\mathcal{F})$ with basic mean curvature that depends on the given non-negative lower bound of the Ricci curvature…

In this paper we prove two backward uniqueness theorems for extrinsic geometric flow of possibly non-compact hypersurfaces in general ambient complete Riemannian manifolds. These are applicable to a wide range of extrinsic geometric flow,…

We consider the graphical mean curvature flow of strictly area decreasing maps $f:M\to N$, where $M$ is a compact Riemannian manifold of dimension $m>1$ and $N$ a complete Riemannian surface of bounded geometry. We prove long-time existence…

We introduce a regularization method for mean curvature flow of a submanifold of arbitrary codimension in the Euclidean space, through higher order equations. We prove that the regularized problems converge to the mean curvature flow for…

We prove the longtime existence for the mean curvature flow problem with a perpendicular Neumann boundary condition in a Generalized Robertson Walker (GRW) spacetime that obeys the null convergence condition. In addition, we prove that the…