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…

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

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

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.

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

Proves smoothness of conical singularities in mean curvature flow.

Proves mean curvature flow from conical singularities to shrinkers.

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.

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

Study of mean curvature flow with obstacles using singular perturbation.

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 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 …

Constructs self-shrinkers with unique asymptotic behavior.

We study a credit risk model which captures effects of economic interactions on a firm's default probability. Economic interactions are represented as a functionally defined graph, and the existence of both cooperative, and competitive, business relations is taken into account. We provide an analytic solution of the mo…

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

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…