arXiv research

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.

Paper proves short-term existence of fractional mean curvature flow.

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 developes a singularity before it can shrink to a point. When $n > 3$ this resul…

We show existence of homothetically shrinking solutions of the fractional mean curvature flow, whose boundary consists in a prescribed numbers of concentric spheres. We prove that all these solutions, except from the ball, are dynamically unstable.

In this paper we study smooth solutions to a fractional mean curvature flow equation. We establish a comparison principle and consequences such as uniqueness and finite extinction time for compact solutions. We also establish evolutions equations for fractional geometric quantities that yield preservation of certain qu…

Smoothness of graphs evolving by fractional mean curvature is proven.

We study a fractional conformal curvature flow on the standard unit sphere and prove a perturbation result of the fractional Nirenberg problem with fractional exponent $σ\in (1/2,1)$. This extends the result of Chen-Xu (Invent. Math. 187, no. 2, 395-506, 2012) for the scalar curvature flow on the standard unit sphere.

First variation of fractional $k$-dimensional measure for submanifolds

Geodesic flows on surfaces have specific fractional-linear integrals related to constant cross-ratios.

We study hypersurfaces of $\mathbb{R}^N$ with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter functional under a volume constraint. We establish the existence of a smooth branch of periodic cylinders in $\mathbb{R}^N$, $N\geq 2$, all of th…

Study examines surfaces with bounded fractional mean curvature, proving control over local parametrization.

We are concerned with hypersurfaces of $\mathbb{R}^N$ with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter under a volume constraint. Our results are twofold. First we prove the nonlocal analogue of the Alexandrov result characterizing sph…

We study fractional configurations in gravity theories and Lagrange mechanics. The approach is based on Caputo fractional derivative which gives zero for actions on constants. We elaborate fractional geometric models of physical interactions and we formulate a method of nonholonomic deformations to other types of fract…

Paper introduces new flows to find circle packings with specific curvature.

Here a new notion of fractional length of a smooth curve, which depends on a parameter $σ$, is introduced that is analogous to the fractional perimeter functional of sets that has been studied in recent years. It is shown that in an appropriate limit the fractional length converges to the traditional notion of length u…

We study the provenance of singularity formation under mean curvature flow and volume preserving mean curvature flow in an axially symmetric setting. We prove that if the mean curvature is uniformly bounded on any finite time interval, then no singularities can develop during that time under both mean curvature flow an…

Study on singularity behavior of mean curvature flow with bounded curvature and index.

Study shows unique tangent flow for Lagrangian surfaces with bounded mean curvature.

Proves uniqueness of geometric flow in various Riemannian manifolds.

Alexandrov immersed surfaces maintain their properties under mean curvature flow.

The paper studies mean curvature flow in a Ricci flow background with extended Ricci flow.

Modeling bone microarchitecture adaptation using geometric flows.

Ancient solutions to mean curvature flow have unique shapes.

Mean curvature flow is not a gradient flow on two nondegenerate metric spaces.

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 case of Finsler manifolds, we will prove the existence and uniqueness for solut…

Mean curvature flow with a conical end becomes stable.

Article explains and implements mean curvature flow for surface parametrization.

Study proves a new flow method for mean curvature with volume change analysis.

In this paper, we investigate the Gauss maps of a Ricci-mean curvature flow. A Ricci-mean curvature flow is a coupled equation of a mean curvature flow and a Ricci flow on the ambient manifold. Ruh and Vilms proved that the Gauss map of a minimal submanifold in a Euclidean space is a harmonic map, and Wang extended thi…

Study examines preservation of curvature-adaptedness during mean curvature flow.

Proves existence and uniqueness of mean curvature flow.

Quantitative estimate for curvature in mean curvature flow.

Proves existence of shrinkers via mean curvature flow.

Smooth convergence to equidistant surfaces in Fuchsian manifolds.

We study graphical mean curvature flow of complete solutions defined on subsets of Euclidean space. We obtain smooth long time existence. The projections of the evolving graphs also solve mean curvature flow. Hence this approach allows to smoothly flow through singularities by studying graphical mean curvature flow wit…

Paper proves Hamilton's pinching theorem using mean curvature flow.

Constructs a mean curvature flow with surgery for compact mean convex hypersurfaces.

Proves Hamilton's theorem using mean curvature flow.

No breathers found for mean curvature flow in noncompact space.

Study shows convergence for mean curvature flow on almost minimal totally real submanifolds.

New study confirms some mean curvature flow solutions have bounded mean curvature.

Ancient symplectic solutions to mean curvature flow are flat.

Proves uniqueness of blowups for forced mean curvature flow.

Ancient solutions found for mean curvature flows in hyperbolic spaces.

Study proves existence and uniqueness of ancient flows from cones.

Paper proves Harnack inequality for $f$-mean curvature flow.

Ancient flows converge fast with finite curvature and convexity.

In this paper, we discuss uniqueness and backward uniqueness for mean curvature flow of non-compact manifolds. We use an energy argument to prove two uniqueness theorems for mean curvature flow with possibly unbounded curvatures. These generalize the results by Chen and Yin. Using similar method, we also obtain a uniqu…