https://neckpinch.com/authors/zhonggan-huang/New arXiv papers by Zhonggan Huang, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2212.03701/arxiv:2212.03701Sun, 04 Dec 2022 00:00:00 GMTZhonggan Huang — It is well-known that the mean curvature flow is a formal gradient flow of the perimeter functional. However, by the work of Michor and Mumford [7,8], the formal Riemannian structure that is compatible with the gradient flow structure induces a degenerate metric on the space of hypersurfaces. It is then natural to ask whether there is a nondegenerate metric space of hypersurfaces, on which the mean curvature flow admits a gradient flow structure. In this paper we study the mean curvature flow on two nondegenerate metric spaces of simple closed plane curves: the uniformness-preserving metric structure proposed by Shi and Vorotnikov [11] and the curvature-weighted structure proposed by Michor and Mumford [8], and prove that the mean curvature flow is not a gradient flow in either of the spaces.Mean curvature flow