Related papers: Avoidance for Set-Theoretic Solutions of Mean-Curv…
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…
In this paper, by considering a special case of the spacelike mean curvature flow investigated by Li and Salavessa [6], we get a condition for the existence of smooth solutions of the Dirichlet problem for the minimal surface equation in…
In this work we study the existence of solutions to the Mean Curvature Flow for which the initial condition has the structure of a two-dimensional Lie subgroup within a Lie group of dimension three. We consider Lie groups with a fixed…
We prove the Multiplicity One Conjecture for mean curvature flows of surfaces in $\mathbb{R}^3$. Specifically, we show that any blow-up limit of such mean curvature flows has multiplicity one. This has several applications. First, combining…
A new monotone quantity in graphical mean curvature flows of higher codimensions is identified in this work. The submanifold deformed by the mean curvature flow is the graph of a map between Riemannian manifolds, and the quantity is…
In this work, we propose a new evolving geometric flow (called translating mean curvature flow) for the translating solitons of hypersurfaces in $R^{n+1}$. We study the basic properties, such as positivity preserving property, of the…
Let $F:\Sigma^n \times [0,T)\to \R^{n+m}$ be a family of compact immersed submanifolds moving by their mean curvature vectors. We show the Gauss maps $\gamma:(\Sigma^n, g_t)\to G(n, m)$ form a harmonic heat flow with respect to the…
We are concerned with the theory of existence and uniqueness of flows generated by divergence free vector fields with compact support. Hence, assuming that the velocity vector fields are measurable, bounded, and the flows in the Euclidean…
In this paper, we study the $k$-cylindrical singular set of mean curvature flow in $\mathbb R^{n+1}$ for each $1\leq k\leq n-1$. We prove that they are locally contained in a $k$-dimensional $C^{2,\alpha}$-submanifold after removing some…
We prove that sufficiently low-entropy closed hypersurfaces can be perturbed so that their mean curvature flow encounters only spherical and cylindrical singularities. Our theorem applies to all closed surfaces in $\mathbb{R}^3$ with…
We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower…
We examine the use of domain decomposition for potentially more efficient mean curvature flow of surface meshes, whose faces are arbitrary simple polygons. We first test traditional domain decomposition methods with and without overlap of…
In this paper, we establish uniform and sharp volume estimates for the singular set and the quantitative singular strata of mean curvature flows starting from a smooth, closed, mean-convex hypersurface in $\mathbb R^{n+1}$.
We study the evolution of hypersurfaces in spacetime initial data sets by their null mean curvature. A theory of weak solutions is developed using the level-set approach. Starting from an arbitrary mean convex, outer untapped hypersurface…
In this article we study the shape of a compact surface of constant mean curvature of Euclidean space whose boundary is contained in a round sphere. We consider the case that the boundary is prescribed or that the surface meets the sphere…
We study a Neumann problem related to the evolution of graphs under mean curvature flow in Riemannian manifolds endowed with a Killing vector field. We prove that in a particular case these graphs converge to a bounded minimal graph which…
The self-similar solutions to the mean curvature flows have been defined and studied on the Euclidean space. In this paper we initiate a general treatment of the self-similar solutions to the mean curvature flows on Riemannian cone…
We propose a level-set method for a mean curvature flow whose boundary is prescribed by interpreting the boundary as an obstacle. Since the corresponding obstacle problem is globally solvable, our method gives a global-in-time level-set…
In this article, we introduce a new type of mean curvature flow for bounded star-shaped domains in space forms and prove its longtime existence, exponential convergence without any curvature assumption. Along this flow, the enclosed volume…
In this paper, we investigate the mean curvature flow of compact surfaces in $4$-dimensional space forms. We prove the convergence theorems for the mean curvature flow under certain pinching conditions involving the normal curvature, which…