相关论文: Construction of High Codimension Ancient Mean Curv…
Several classes of solutions of the generalized Weierstrass system, which induces constant mean curvature surfaces into four-dimensional Euclidean space are constructed. A gauge transformation allows us to simplify the system considered and…
I classify spacelike self-similar shrinking solutions of the mean curvature flow in pseudo-euclidean space in arbitrary codimension, if the mean curvature vector is not a null vector and the principal normal vector is parallel in the normal…
We establish short-time existence of the smooth solution to the fractional mean curvature flow when the initial set is bounded and C^{1,1}-regular. We provide the same result also for the volume preserving fractional mean curvature flow.
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…
We show that flatness of the normal bundle is preserved under the mean curvature flow in the Euclidean space and use this to generalize a classical result for hypersurfaces due to Ecker-Huisken in the case of submanifolds with arbitrary…
In this paper, we study self-expanding solutions for mean curvature flows and their relationship to minimal cones in Euclidean space. In [18], Ilmanen proved the existence of self-expanding hypersurfaces with prescribed tangent cones at…
We prove a sharp pinching estimate for immersed mean convex solutions of mean curvature flow which unifies and improves all previously known pinching estimates, including the umbilic estimate of Huisken, the convexity estimates of…
This paper is concerned with the mean curvature flow, which describes the dynamics of a hypersurface whose normal velocity is determined by local mean curvature. We present a Cartesian grid-based method for solving mean curvature flows in…
For hypersurfaces of dimension greater than one, Huisken showed that compact self-shrinkers of the mean curvature flow with positive scalar mean curvature are spheres. We will prove the following extension: A compact self-similar solution…
Huisken studied asymptotic behavior of a mean curvature flow in a Euclidean space when it develops a singularity of type I, and proved that its rescaled flow converges to a self-shrinker in the Euclidean space. In this paper, we generalize…
In this paper, we first investigate the integral curvature condition to extend the mean curvature flow of submanifolds in a Riemannian manifold with codimension $d\geq1$, which generalizes the extension theorem for the mean curvature flow…
In this paper we refine the construction and related estimates for complete Constant Mean Curvature surfaces in Euclidean three-space developed in Kapouleas (1990) by adopting the more precise and powerful version of the methodology which…
In this paper we introduce the hyperbolic mean curvature flow and prove that the corresponding system of partial differential equations are strictly hyperbolic, and based on this, we show that this flow admits a unique short-time smooth…
We prove that smooth convex $\alpha$-noncollapsed ancient mean curvature flow satisfies a quantitative curvature estimate $H(y,t)\leq CH(x,t)(H(x,t)|x-y|+1)^2$ for any pair of $x,y$. In other words, the rescaled curvature grows at most…
We consider ancient solutions to the mean curvature flow in $\mathbb{R}^{n+1}$ ($n \geq 3$) that are weakly convex, uniformly two-convex, and satisfy derivative estimates $|\nabla A| \leq \gamma_1 |H|^2, |\nabla^2 A| \leq \gamma_2 |H|^3$.…
In this paper, we study the mean curvature type flow for hypersurfaces in the unit Euclidean ball with capillary boundary, which was introduced by Wang-Xia and Wang-Weng. We show that if the initial hypersurface is strictly convex, then the…
We provide explicit examples which show that mean convexity (i.e. positivity of the mean curvature) and positivity of the scalar curvature are non-preserved curvature conditions for hypersurfaces of the Euclidean space evolving under either…
We consider the evolution of a $n$-dimensional convex hypersurface in the euclidean space under mean curvature flow with densities $e^{\varepsilon \frac12 n\mu^2 |x|^2}$, $\varepsilon =\pm 1$, and completely determine it depending on the…
Mean curvature flow evolves isometrically immersed base manifolds $M$ in the direction of their mean curvatures in an ambient manifold $\bar{M}$. If the base manifold $M$ is compact, the short time existence and uniqueness of the mean…
We consider the inverse mean curvature flow by parallel hypersurfaces in space forms. We show that such a flow exists if and only if the initial hypersurface is isoparametric. The flow is characterized by an algebraic equation satisfied by…