As on arXiv
Abstract
We study the dynamics of planar sets under area-preserving crystalline curvature flow. We prove under mild assumptions on 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 -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 -minimal barriers for energy competition arguments, both for the geometric rigidity of the discretized flat flow and for the regularity of almost-minimizers.
Comments: 42 pages, 6 figures
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