https://neckpinch.com/authors/eric-kim/New arXiv papers by Eric Kim, from Neckpinch Tue, 03 Nov 2026 07:43:57 GMT https://neckpinch.com/papers/2606.06911/arxiv:2606.06911Fri, 05 Jun 2026 00:00:00 GMTEric Kim — We study the dynamics of planar sets under area-preserving crystalline curvature flow. We prove under mild assumptions on φ that the flat flow solution from regular initial data coincides with a classical ODE evolution, extending the results of to the area-preserving setting. We also show that for arbitrary initial data, the flat flow converges exponentially in time to a disjoint union of Wulff shapes, and under a non-bubbling assumption, the flow eventually becomes regular. Both of these results are novel for area-preserving crystalline flow of general sets, i.e. without assuming geometric properties such as convexity or star-shapedness. A key ingredient of independent interest is that planar almost-minimizers are Lipschitz φ-regular, which we prove by exploiting a sharp minimality estimate for distinguished line segments, as opposed to the excess decay argument given in. The novelty of our approach lies in the application of φ-minimal barriers for energy competition arguments, both for the geometric rigidity of the discretized flat flow and for the regularity of almost-minimizers.Other curvature flowsVolume-preserving & constrained flowsAnisotropic & nonlocal flowsStability & uniquenessLong-time existence & convergence https://neckpinch.com/papers/2405.08296/arxiv:2405.08296Tue, 14 May 2024 00:00:00 GMTEric Kim, Dohyun Kwon — We study the motion of sets by anisotropic curvature under a volume constraint in the plane. We establish the exponential convergence of the area-preserving anisotropic flat flow to a disjoint union of Wulff shapes of equal area, the critical point of the anisotropic perimeter functional. This is an anisotropic analogue of the results in the isotropic case studied in. The novelty of our approach is in using the Cahn-Hoffman map to parametrize boundary components as small perturbations of the Wulff shape. In addition, we show that certain reflection comparison symmetries are preserved by the flat flow, which lets us obtain uniform bounds on the distance between the convergent profile and the initial data.Mean curvature flowVolume-preserving & constrained flowsAnisotropic & nonlocal flowsLong-time existence & convergence