Related papers: The curve shortening problem associated to a densi…
The paper presents the results of numerical solution of the evolution law for the constrained mean-curvature flow. This law originates in the theory of phase transitions for crystalline materials and describes the evolution of closed…
In this paper we introduce the target flow -- a specific curve shortening flow with an ambient forcing term -- that, given an embedded (not necessarily convex) target curve, will attempt to evolve a given source curve to that target. The…
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 consider the flow of closed convex hypersurfaces in Euclidean space $\mathbb{R}^{n+1}$ with speed given by a power of the $k$-th mean curvature $E_k$ plus a global term chosen to impose a constraint involving the enclosed volume…
Let N be a (n+1)-dimensional globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface. We consider curvature flows in N with different curvature functions F (including the mean curvature, the gauss curvature and the second…
In a rotationally symmetric space $\oM$ around an axis A (whose precise definition includes all real space forms), we consider a domain $G$ limited by two equidistant hypersurfaces orthogonal to A. Let $M \subset \oM$ be a revolution…
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…
Studying the geometric flow plays a powerful role in mathematics and physics. In this paper, we introduce the mean curvature flow on Finsler manifolds and give a number of examples of the mean curvature flow. For Minkowski spaces, a special…
We establish existence and uniqueness results for the modified binormal curvature flow equation that generalizes the binormal curvature flow equation for a curve in $\R^3.$ In this generalization, the velocity of the curve is still directed…
In this paper, we consider the mean curvature flow with driving force on fixed extreme points in the plane. We give a general local existence and uniqueness result of this problem with $C^2$ initial curve. For a special family of initial…
We introduce a simple and efficient numerical method to compute mean curvature flow with obstacles. The method augments the Merrimam-Bence-Osher scheme with a pointwise update that enforces the constraint and therefore retains the…
This paper concerns closed hypersurfaces of dimension $n(\geq 2)$ in the hyperbolic space ${\mathbb{H}}_{\kappa}^{n+1}$ of constant sectional curvature $\kappa$ evolving in direction of its normal vector, where the speed is given by a power…
We define a notion of mean curvature flow with surgery for two-dimensional surfaces in $\mathbb{R}^3$ with positive mean curvature. Our construction relies on the earlier work of Huisken and Sinestrari in the higher dimensional case. One of…
Our aim is to study invariant hypersurfaces immersed in the Euclidean space $\mathbb{R}^{n+1}$, whose mean curvature is given as a linear function in the unit sphere $\mathbb{S}^n$ depending on its Gauss map. These hypersurfaces are closely…
This paper gives some examples of hypersurfaces $\phi_t(M^n)$ evolving in time with speed determined by functions of the normal curvatures in an $(n+1)$-dimensional hyperbolic manifold; we emphasize the case of flow by harmonic mean…
We consider the mean curvature flow of the graph of a smooth map $f:\mathbb{R}^2\to\mathbb{R}^2$ between two-dimensional Euclidean spaces. If $f$ satisfies an area-decreasing property, the solution exists for all times and the evolving…
We consider curve shortening flow of arbitrary codimension in an Euclidean background. We show that, close to a singularity, the flow is asymptotically planar, paralleling Altschuler's work in the case of space curves, and analyse the…
A variant of the Gauss curvature flow for closed and convex hypersurfaces is considered. We reveal that if the initial hypersurface is pinched enough, then this property is preserved. Furthermore, based on some structure assumptions on the…
In this paper we study isentropic flow in a curved pipe. We focus on the consequences of the geometry of the pipe on the dynamics of the flow. More precisely, we present the solution of the general Cauchy problem for isentropic fluid flow…
In this note, we discuss the mean curvature flow of graphs of maps between Riemannian manifolds. Special emphasis will be placed on estimates of the flow as a non-linear parabolic system of differential equations. Several global existence…