Following & Lstrok;ojasiewicz's uniqueness theorem and Thom's gradient conjecture, Arnold proposed a stronger version about the existence of limit tangents of gradient flow lines for analytic functions. We prove & Lstrok;ojasiewicz's theorem and Arnold's conjecture in the context of arrival time functions for mean curvature flows in & Ropf;nC1 with neck or nondegenerate cylindrical singularities. In particular, we prove the conjectures for all mean convex mean curvature flows of surfaces, including the cases when the arrival time functions are not C2. The results also apply to mean curvature flows starting from two-spheres or generic closed surfaces.

We prove a parabolically scale-invariant variation of the planarity estimate in [Naff, K.: A planarity estimate for pinched solutions of mean curvature flow. Duke Math. J. 171(2), 443–482 (2022) ] for higher codimension mean curvature flow, borrowing ideas from work of Brendle–Huisken–Sinestrari [Brendle, S., Huisken, G., Sinestrari, C.: Ancient solutions to the Ricci flow with pinched curvature. Duke Math. J. 158(3), 537–551 (2011) ]. Additionally, we prove convexity for pinched complete ancient solutions of the mean curvature flow in codimension one. Then we put these estimates together to characterize certain pinched complete ancient solutions and shrinkers in higher codimension. We include some discussion of future research directions in this area of mean curvature flow.

We study ancient solutions to discrete heat equations on some weighted graphs. On a graph of the form of a product with Z, we show that there are no non-trivial ancient solutions with polynomial growth. This result is parallel to the case of finite graphs, which is also discussed. Along the way, we prove a backward uniqueness result for solutions with appropriate decaying rate based on a monotonicity formula of parabolic frequency.

In this paper, we prove that for any asymptotically conical self-shrinker, there exists a closed hypersurface such that the mean curvature flow starting from it develops a singularity modeled on the given shrinker. The main technique is the Ważewski box argument, used by Stolarski in the proof of the corresponding theorem in the Ricci flow case.

In this paper, we prove the uniqueness of asymptotically conical tangent flows in all codimensions. This is based on an early work of Chodosh-Schulze, who proved the uniqueness in the hypersurface case.

This paper defines a parabolic frequency for solutions of the heat equation along homothetically shrinking mean curvature flows (MCFs) and proves its monotonicity along such flows. As a corollary, frequency monotonicity provides a proof of backwards uniqueness. Additionally, for solutions of more general parabolic equations on MCF shrinkers, this paper provides bounds on the derivative of the frequency, which similarly imply backwards uniqueness.

This paper defines a parabolic frequency for solutions of the heat equation along homothetically shrinking mean curvature flows and proves its monotonicity along such flows. As a corollary, frequency monotonicity provides a proof of backwards uniqueness. Additionally, for solutions of more general parabolic equations on mean curvature flow shrinkers, this paper provides bounds on the derivative of the frequency, which similarly imply backwards uniqueness.

The entropy functional introduced by Colding and Minicozzi plays a fundamental role in the analysis of mean curvature flow. However, unlike the hypersurface case, relatively little about the entropy is known in the higher-codimension case. In this note, we use measure-theoretical techniques and rigidity results for self-shrinkers to prove a compactness theorem for a family of self-shrinking surfaces with low entropy. Based on this, we prove the existence of entropy minimizers among self-shrinking surfaces and improve some rigidity results.

We prove that any n n -dimensional closed mean convex λ \lambda - hypersurface is convex if λ ≤ 0. \lambda \le 0. This generalizes Guang’s work on 2 2 -dimensional strictly mean convex λ \lambda -hypersurfaces. As a corollary, we obtain a gap theorem for closed λ \lambda -hypersurfaces with λ ≤ 0. \lambda \le 0.

The entropy functional introduced by Colding and Minicozzi plays a fundamental role in the analysis of mean curvature flow. However, unlike the hypersurface case, relatively little about the entropy is known in the higher-codimension case. In this note, we use measure-theoretical techniques and rigidity results for self-shrinkers to prove a compactness theorem for a family of self-shrinkers with low entropy. Based on this, we prove that there exists a $2$-dimensional complete, non-flat, and smooth embedded self-shrinker that minimizes the entropy among all $2$-dimensional non-flat self-shrinkers in $\mathbb R^N$ for any $N.$