Related papers: Gauss curvature flow with shrinking obstacle
We develop a global theory for complete hypersurfaces in $\mathbb{R}^{n+1}$ whose mean curvature is given as a prescribed function of its Gauss map. This theory extends the usual one of constant mean curvature hypersurfaces in…
We prove that for the mean curvature flow of closed embedded hypersurfaces, the intrinsic diameter stays uniformly bounded as the flow approaches the first singular time, provided all singularities are of neck or conical type. In…
This article discusses a relatively new geometric flow, called the hypersymplectic flow. In the first half of the article we explain the original motivating ideas for the flow, coming from both 4-dimensional symplectic topology and…
In this paper we look for the convex hull of a set using the geometric evolution by minimal curvature of a hypersurface that surrounds the set. To find the convex hull, we study the large time behavior of solutions to an obstacle problem…
An existence and uniqueness result, up to fattening, for crystalline mean curvature flows with forcing and arbitrary (convex) mobilities, is proven. This is achieved by introducing a new notion of solution to the corresponding level set…
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…
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.
We discuss the property of the number density of a fluid of particles living in a curved surface without boundaries to be constant in the thermodynamic limit. In particular we find a sufficient condition for the density to be constant along…
In this paper, we consider a new length preserving curve flow for convex curves in the plane. We show that the global flow exists, the area of the region bounded by the evolving curve is increasing, and the evolving curve converges to the…
In this paper, we study the power of Gaussian curvature flow of a compact convex hypersurface and establish its Harnack inequality when the power is negative. In the Harnack inequality, we require that the absolute value of the power is…
The conformal flow of metrics [2] has been used to successfully establish a special case of the Penrose inequality, which yields a lower bound for the total mass of a spacetime in terms of horizon area. Here we show how to adapt the…
Ensuring a smooth rate of efflux of particles from an outlet without unpredictable clogging events is crucial in processing powders and grains. We show by experiments and simulations that an obstacle placed near the outlet can greatly…
We prove the existence of a weak global in time mean curvature flow of a bounded partition of space using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We also prove some…
We consider the rescaled flow associated with a mean curvature flow that develops a compact singularity of multiplicity one. We prove that the ``decay order'' of such a rescaled flow is uniformly bounded. As a consequence, we prove a unique…
In this paper, we study the deformation of the 2 dimensional convex surfaces in $\R^{3}$ whose speed at a point on the surface is proportional to $\alpha$-power of positive part of Gauss Curvature. First, for 1/2<\alpha\leq 1$, we show that…
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…
We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find pinching conditions on the second fundamental form that characterize the shrinking sphere among compact ancient solutions…
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 prove the asymptotic roundness under normalized Gauss curvature flow provided entropy is initially small enough.
We show that for certain one-parameter families of initial conditions in $\mathbb R^3$, when we run mean curvature flow, a genus one singularity must appear in one of the flows. Moreover, such a singularity is robust under perturbation of…