arXiv research

A mean-convex set can be regarded as a barrier for the construction of minimal surfaces. Namely, if we are given a mean-convex set and a null-homotopic Jordan curve on its boundary, then there exists an embedded minimal disk with boundary the given curve contained in the starting mean-convex set. Does a mean-convex set…

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

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…

This note generalizes the visual angle to convex sets 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.

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

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…

Study on estimating Gaussian mean from coarse data, resolving identifiability and computational efficiency questions.

The paper proves inequalities for star-shaped and $F$-mean convex hypersurfaces in $\mathbb{R}^{n+1}$.

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…

Proves Riemannian positive mass theorem with singularities.

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…

New principles for collapsing law-invariant functionals to means, extending beyond convexity.

The paper studies stability and singularities of a two-convex level set flow.

We obtain estimates on both size and dimensions of the singular set at the first blow-up time of the mean curvature flow of hypersurfaces whose initial data is $σ_k$-convex.

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…

We obtain a sharp characterization of the Euclidean ball among all convex bodies K whose boundary has a pointwise k-th mean curvature not smaller than a geometric constant at almost all normal points. This geometric constant depends only on the volume and the boundary area of K. We deduce this characterization from a n…

The study proves splitting theorems for manifolds with specific curvature and boundary conditions.

Study shows bound on Uryson width for specific 3D manifolds.

Convexity preserved in curved surfaces moving at concave speeds.

Consider the mean curvature flow of an (n+1)-dimensional, compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the m-th homotopy group of the complementary region can die only if there is a shrinking S^k x R^(n-k) singularity for some k less than or equal to m…

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.

The study finds at least 2 free-boundary minimal disks in convex 3-balls.

Sharp upper bounds derived for capacities in hyperbolic and Euclidean spaces.

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

We show compactness in the locally smooth topology for certain natural families of asymptotically conical self-expanding solutions of mean curvature flow. Specifically, we show such compactness for the set of all two-dimensional self-expanders of a fixed topological type and, in all dimensions, for the set of self-expa…

Self-shrinkers model singularities of the mean curvature flow; they are defined as the special solutions that contract homothetically under the flow. Colding-Ilmanen-Minicozzi showed that cylindrical self-shrinkers are rigid in a strong sense - that is, any self-shrinker that is mean convex with uniformly bounded curva…

The paper estimates the volume of singular points in evolving surfaces.

The paper explores connections between perimeter, area, and visual angle of convex sets.

Sharp upper bounds on inscribed radius for metric spaces with convex boundary.

Paper characterizes monotonic mean-deviation risk measures.

New method proves existence of constant mean curvature disks on convex surfaces.

The paper analyzes the mean field Langevin dynamics and its convergence rate.

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…

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

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

Resolves conjecture on cylindrical mean curvature flows in all dimensions.

Unbounded convex domains have zero mean curvature on disconnected boundaries.

We study properly immersed ancient solutions of the codimension one mean curvature flow in $n$-dimensional Euclidean space, and classify the convex hulls of the subsets of space reached by any such flow. In particular, it follows that any compact convex ancient mean curvature flow can only have a slab, a halfspace or a…

A special formula for the total mean curvature of an ovaloid is derived. This formula allows us to extend the notion of the mean curvature to the class of boundaries of strictly convex sets. Moreover, some integral formula for ovaloids is proved.

We solve a portfolio selection problem with four objectives, finding convex scalarizations for part of the Pareto front.

We consider $n$-dimensional hypersurfaces flowing by mean curvature flow with Neumann free boundary conditions supported on a smooth support surface. We show that the Hausdorff $n$-measure of the singular set is zero. In fact, we consider two types of interaction between the support and flowing surfaces. In the case of…

Sharp bounds for anisotropic p-capacity of Euclidean compact sets derived using flow methods.

Alexandrov immersed surfaces maintain their properties under mean curvature flow.

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

Smooths out complex shapes into simpler forms.

Proves convexity of certain hypersurfaces with negative λ.

The study extends curvature bounds to non-smooth spaces and proves stability of mean curvature.