arXiv research

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

Strict convexity proven for certain self-expanders in high dimensions.

The study characterizes and rules out collapsing in convex ancient mean curvature flow.

Proves flows of two-convex Lagrangians are regular, global, and converge.

Unique ancient convex flow in a ball with free boundary found.

Ancient Lagrangian flows get limited convex solutions.

New proof shows symmetry for certain curved surfaces in higher dimensions.

Proves planarity and convexity for ancient solutions of mean curvature flow.

Study shows thresholding scheme converges for mean curvature flow of convex sets.

Paper relaxes convexity assumptions in mean curvature flow results.

High codimension submanifolds evolve to convex shapes, leading to smooth limiting flows.

We prove the convexity estimates of Huisken-Sinestrari for finite-time singularities of mean-convex, mean curvature flow with free boundary in a barrier $S$. Here $S$ can be any properly embedded, oriented surface in $R^{n+1}$ of bounded geometry. We also give an alternative proof that convex mean curvature flows with …

We prove that for the mean curvature flow of two-convex hypersurfaces the intrinsic diameter stays uniformly controlled as one approaches the first singular time. We also derive sharp $L^{n-1}$-estimates for the regularity scale of the level set flow with two-convex initial data. Our proof relies on a detailed analysis…

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…

We show that a mean curvature flow starting from a compact, smoothly embedded hypersurface M remains unique past singularities, provided the singularities are of mean convex type, i.e., if around each singular point, the surface moves in one direction. Specifically, the level set flow of M does not fatten if all singul…

The paper proves convexity of certain solitons and expanders in high dimensions.

The study classifies flows of finite curvature in 3D space.

In this note we announce results on the mean curvature flow of mean convex sets in 3-dimensions. Loosely speaking, our results justify the naive picture of mean curvature flow where the only singularities are neck pinches, and components which collapse to asymptotically round spheres.

We show that any strictly mean convex translator of dimension $n\geq 3$ which admits a cylindrical estimate and a corresponding gradient estimate is rotationally symmetric. As a consequence, we deduce that any translating solution of the mean curvature flow which arises as a blow-up limit of a two-convex mean curvature…

Resolves conjecture on cylindrical mean curvature flows in all dimensions.

Ancient convex solutions to flow equations are limited to simple shapes.

Study ancient solutions to free boundary mean curvature flow in convex manifolds.

We consider the mean curvature flow of compact convex surfaces in Euclidean $3$-space with free boundary lying on an arbitrary convex barrier surface with bounded geometry. When the initial surface is sufficiently convex, depending only on the geometry of the barrier, the flow contracts the surface to a point in finite…

A possible evolution of a compact hypersurface in R^n by mean curvature past singularities is defined via the level set flow. In the case that the initial hypersurface has positive mean curvature, we show that the Brakke flow associated to the level set flow is actually a Brakke flow with equality. We obtain as a conse…

New Harnack inequality for curve shortening flow without convexity.

In the last 15 years, White and Huisken-Sinestrari developed a far-reaching structure theory for the mean curvature flow of mean convex hypersurfaces. Their papers provide a package of estimates and structural results that yield a precise description of singularities and of high curvature regions in a mean convex flow.…

Study proves rigidity theorems for ancient solutions to mean curvature flow with convex image.

The paper studies a flow for convex capillary hypersurfaces in a ball, proving smooth convergence to a spherical cap.

In this article, we will use the harmonic mean curvature flow to prove a new class of Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces in hyperbolic space in terms of total curvature, which is the integral of Gaussian curvature on the hypersurface. We will also use the harmonic mean curvature flow…

Quantitative estimate for curvature in mean curvature flow.

Constructs ancient solutions to mean curvature flow with prescribed singular sets.

For any $n$-dimensional smooth manifold $Σ$, we show that all the singularities of the mean curvature flow with any initial mean convex hypersurface in $Σ$ are cylindrical (of convex type) if the flow converges to a smooth hypersurface $M_{\infty}$ (maybe empty) at infinity. Previously this was shown (i) for $n\leq 7$,…

Proves existence of mean curvature flow with surgery for free boundary surfaces.

In this paper, we generalize White's regularity and structure theory for mean-convex mean curvature flow to the setting with free boundary. A major new challenge in the free boundary setting is to derive an a priori bound for the ratio between the norm of the second fundamental form and the mean curvature. We establish…

Convexity preserved in curved surfaces moving at concave speeds.

We consider the evolution of hypersurfaces on the unit sphere $\mathbb{S}^{n+1}$ by their mean curvature. We prove a differential Harnack inequality for any weakly convex solution to the mean curvature flow. As an application, by applying an Aleksandrov reflection argument, we classify convex, ancient solutions of the …

We use Ilmanen's elliptic regularization to prove that for an initially smooth mean convex hypersurface in Euclidean n-space moving by mean curvature flow, the surface is very nearly convex in a spacetime neighborhood of every singularity. Previously this was known only (i) for n < 7, and (ii) for arbitrary n up to the…

Ancient flows converge fast with finite curvature and convexity.

In this paper we study the geometry of first time singularities of the mean curvature flow. By the curvature pinching estimate of Huisken and Sinestrari, we prove that a mean curvature flow of hypersurfaces in the Euclidean space $\R^{n+1}$ with positive mean curvature is $κ$-noncollapsing, and a blow-up sequence conve…

In this article, we extend the mean curvature flow with surgery to mean convex hypersurfaces with entropy less than $Λ_{n-2}$. In particular, 2-convexity is not assumed. Next we show the surgery flow with just the initial convexity assumption $H - \frac{\langle x, ν\rangle}{2} > 0$ is possible and as an application we …

New convex ancient solutions found for flows by high powers of curvature.

We prove two new estimates for the level set flow of mean convex domains in Riemannian manifolds. Our estimates give control - exponential in time - for the infimum of the mean curvature, and the ratio between the norm of the second fundamental form and the mean curvature. In particular, the estimates remove a stumblin…

We study inverse mean curvature flows of starshaped, mean convex hypersurfaces in warped product manifolds with a positive warping factor $\varphi(r)$. If $\varphi'(r)>0$ and $\varphi''(r)\geq 0$, we show that these flows exist for all times, remain starshaped and mean convex. Plus the positivity of $\varphi''(r)$ and …

The paper studies mean curvature flow of spacelike-convex submanifolds in pseudo-Euclidean space.

Study self-expanding solutions of mean curvature flow in various dimensions.

The study examines mass drop and multiplicity in mean curvature flow.

Under mean curvature flow, a closed, embedded hypersurface $M(t)$ becomes singular in finite time. For certain classes of mean-convex mean curvature flows, we show the continuity of the first singular time $T$ and the limit set "$M(T)$", with respect to initial data. We employ an Angenent-like neck-pinching argument to…

We give a new proof for the existence of mean curvature flow with surgery of 2-convex hypersurfaces in $R^N$, as announced in arXiv:1304.0926. Our proof works for all $N \geq 3$, including mean convex surfaces in $R^3$. We also derive a priori estimates for a more general class of flows in a local and flexible setting.