arXiv research

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…

Proves smoothness of conical singularities in mean curvature flow.

We consider an axisymmetric closed hypersurface evolving by its mean curvature with driving force under singular initial hypersurface. We study this problem by level set method. We give some criteria to judge whether the interface evolution is fattening or non-fattening.

Proves mean curvature flow from conical singularities to shrinkers.

New non-canonical flows found via parabolic Allen-Cahn equations.

We show that if $Σ\subset \mathbb R^4$ is a closed, connected hypersurface with entropy $λ(Σ)\leq λ(\mathbb{S}^2\times \mathbb R)$, then the level set flow of $Σ$ never disconnects. We also obtain a sharp version of the forward clearing out lemma for non-fattening flows in $\mathbb R^4$ of low entropy.

Study of mean curvature flow with obstacles using singular perturbation.

Mean curvature flow shows a surface fattening at its first singular point.

Researchers construct translators asymptotic to self-shrinkers, proving non-uniqueness and fattening.

In this paper, we prove short time existence and uniqueness of smooth evolution by mean curvature in $\mathbb{R}^{n+1}$ starting from any $n$-dimensional $(\varepsilon,R)$-Reifenberg flat set with $\varepsilon$ sufficiently small. More precisely, we show that the level set flow in such a situation is non-fattening and …

The paper extends the avoidance principle for mean curvature flows, proving new intersection dimension monotonicity results.

We study the existence and uniqueness of smooth mean curvature flow, in arbitrary dimension and co-dimension, emanating from so called $k$-dimensional $(\varepsilon,R)$ Reifenberg flat sets in $\mathbb{R}^n$. Our results generalize the ones from a previous paper by the author, in which the co-dimension one case (i.e. $…

In this paper we introduce a geometric quantity, the $r$-multiplicity, that controls the length of a smooth curve as it evolves by curve shortening flow. The length estimates we obtain are used to prove results about the level set flow in the plane. If $K$ is locally-connected, connected and compact, then the level set…

Proves multiplicity one conjecture for surface flows, with applications in flow regularity.

Constructs self-shrinkers with unique asymptotic behavior.

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

Paper studies generic dynamics of MCFs with spherical singularities.

In this article we use the mean curvature flow with surgery to derive regularity estimates for the level set flow going past Brakke regularity in certain special conditions allowing for 2-convex regions of high density. We also show a stability result for the plane under the level set flow.

Level set flow's singularities are type I under 2-convexity, leading to specific curvature blow-up rates.

In this note we prove that the level-set flow of the topologist's sine curve is a smooth closed curve. In previous work it was shown by the second author that under level-set flow, a locally-connected set in the plane evolves to be smooth, either as a curve or as a positive area region bounded by smooth curves. Here we…

Study shows continuous evolution of curves in Fréchet distance.

Families of hypersurfaces that are level-set families of harmonic functions free of critical points are characterized by a local differential-geometric condition. Harmonic functions with a specified level-set family are constructed from geometric data. As a by-product, it is shown that the evolution of the gradient of …

Generic level sets in mean curvature flow are BV solutions.

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…

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…

The paper connects two clustering methods by showing gradient ascent flow can move up the cluster tree.

We showed earlier that the level set function of a monotonic advancing front is twice differentiable everywhere with bounded second derivative. We show here that the second derivative is continuous if and only if the flow has a single singular time where it becomes extinct and the singular set consists of a closed $C^1…

In this paper, we first discuss the regular level set of a nonsingular Smale flow (NSF) on a 3-manifold. The main result about this topic is that a 3-manifold $M$ admits an NSF flow which has a regular level set homeomorphic to $(n+1)T^{2}$ $(n\in \mathbb{Z}, n\geq 0)$ if and only if $M=M'\sharp n S^{1}\times S^{2}$. T…

Continuous curve evolution depends on initial shape on sphere.

The paper characterizes potential functions whose level sets are orbits in mechanical systems.

We study the multi-level order-flow imbalance (MLOFI), which is a vector quantity that measures the net flow of buy and sell orders at different price levels in a limit order book (LOB). Using a recent, high-quality data set for 6 liquid stocks on Nasdaq, we fit a simple, linear relationship between MLOFI and the conte…

In this paper we prove that if $γ$ is a Jordan curve on $\mathbb{S}^2$ then there is a smooth curve shortening flow defined on $(0,T)$ which converges to $γ$ in $\mathcal{C}^0$ as $t\to 0^+$. Another perspective is that the level-set flow of $γ$ is smooth. This is a generalization of the author's previous work where t…

The study uses heat flow to analyze properties of Laplace eigenfunctions on manifolds and domains.

Unified view of monotonicity formulas for inverse mean curvature flow and $p$-capacitary potentials.

For Hamiltonian flows we establish the existence of periodic orbits on a sequence of level sets approaching a Bott-nondegenerate symplectic extremum of the Hamiltonian. As a consequence, we show that a charge on a compact manifold with a nondegenerate (i.e. symplectic) magnetic field has periodic orbits on a sequence o…

In this paper we study the generalized mean curvature flow of sets in the sub-Riemannian geometry of Carnot groups. We extend to our context the level sets method and the weak (viscosity) solutions introduced in the Euclidean setting by Evans-Spruck and Chen-Giga-Goto. We establish two special cases of the comparison p…

Hamiltonian dynamical systems tend to have infinitely many periodic orbits. For example, for a broad class of symplectic manifolds almost all levels of a proper smooth Hamiltonian carry periodic orbits. The Hamiltonian Seifert conjecture is the existence problem for regular compact energy levels without periodic orbits…

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

Study of intersections in Hamiltonian orbits on cotangent bundles.

In this paper, we study the motion of level sets by general curvature. The difficulty of this setting is that a general curvature function is only well defined in an admissible cone. In order to extend the existence of a weak solution of a general curvature flow to outside the cone we introduce a new approximation func…

In this paper we study flows $\varphi:M\times\mathbb{R}\longrightarrow M$ having an isolated non-saddle set. We see that the complexity of the region of influence of an isolated non-saddle set $K$ depends on the way in which $K$ sits on the phase space at the cohomological level. We construct flows in surfaces having i…

The paper constructs hypersurfaces translating under powers of Gauss curvature.

Basic aspects of the equiaffine geometry of level sets are developed systematically. As an application there are constructed families of $2n$-dimensional nondegenerate hypersurfaces ruled by $n$-planes, having equiaffine mean curvature zero, and solving the affine normal flow. Each carries a symplectic structure with r…

In this paper, we prove an extended version of the Minkowski Inequality, holding for any smooth bounded set $Ω\subset \mathbb R^n$, $n\geq 3$. Our proof relies on the discovery of effective monotonicity formulas holding along the level set flow of the $p$-capacitary potentials associated with $Ω$, for every $p$ suffici…

Let $A \subset \mathbb{R}^d$, $d\ge 2$, be a compact convex set and let $μ= \varrho_0 dx$ be a probability measure on $A$ equivalent to the restriction of Lebesgue measure. Let $ν= \varrho_1 dx$ be a probability measure on $B_r := \{x\colon |x| \le r\}$ equivalent to the restriction of Lebesgue measure. We prove that t…

We present a simple connection between differential Harnack inequalities for hypersurface flows and natural concavity properties of their time-of-arrival functions. We prove these concavity properties directly for a large class of flows by applying a concavity maximum principle argument to the corresponding level set f…

Study shows integrating OFI from multiple levels improves price impact explanation but not forecasting.

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…