Related papers: Mean Curvature Flow with Boundary

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…

We survey some of the state of the art regarding singularities in Lagrangian mean curvature flow. Some open problems are suggested at the end.

We provide a direct proof of a non-collapsing estimate for compact hypersurfaces with positive mean curvature moving under the mean curvature flow: Precisely, if every point on the initial hypersurface admits an interior sphere with radius…

We review some recent results on the mean curvature flows of Lagrangian submanifolds from the perspective of geometric partial differential equations. These include global existence and convergence results, characterizations of first-time…

We consider a multiphase surface $\mathcal{C}_0$ in $\mathbb{R}^3$ consisting of a finite number of surfaces passing through the origin , where all 1-dimensional junctions are regular triple junctions in which three planes meet at the same…

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…

In this paper, we mainly study the mean curvature flow in K\"ahler surfaces with positive holomorphic sectional curvatures. We prove that if the ratio of the maximum and the minimum of the holomorphic sectional curvatures is less than 2,…

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 compute the averages over elliptic curves of the constants occurring in the Lang-Trotter conjecture, the Koblitz conjecture, and the cyclicity conjecture. The results obtained confirm the consistency of these conjectures with the…

We give a survey of various existence results for minimal Lagrangian graphs. We also discuss the mean curvature flow for Lagrangian graphs.

We investigate the existence, convergence and uniqueness of modified general curvature flow of convex hypersurfaces in hyperbolic space with a prescribed asymptotic boundary.

We give a proof of the fact that the upper and the lower sectional curvature bounds of a complete manifold vary at a bounded rate under the Ricci flow.

In this paper we investigate the convergence for the mean curvature flow of closed submanifolds with arbitrary codimension in space forms. Particularly, we prove that the mean curvature flow deforms a closed submanifold satisfying a…

In this paper we study the mean curvature flow of embedded disks with free boundary on an embedded cylinder or generalised cone of revolution, called the support hypersurface. We determine regions of the interior of the support hypersurface…

We consider the volume constrained fractional mean curvature flow of a nearly spherical set, and prove long time existence and asymptotic convergence to a ball. The result applies in particular to convex initial data, under the assumption…

As first noted in Korevaar, Kusner and Solomon ("KKS"), constant mean curvature implies a homological conservation law for hypersurfaces in ambient spaces with Killing fields.In Theorem 3.5 here, we generalize that law by relaxing the…

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…

Bounds of total curvature and entropy are two common conditions placed on mean curvature flows. We show that these two hypotheses are equivalent for the class of ancient complete embedded smooth planar curve shortening flows, which are…

We construct maximal hypersurfaces with a Neumann boundary condition in Minkowski space via mean curvature flow. In doing this we give general conditions for long time existence of the flow with boundary conditions with assumptions on the…

We introduce a new notion of viscosity solutions for the level set formulation of the motion by crystalline mean curvature in three dimensions. The solutions satisfy the comparison principle, stability with respect to an approximation by…