Uniformly Compressing Mean Curvature Flow

Wenhui Shi, Dmitry Vorotnikov

Research output: Contribution to journalArticleResearchpeer-review

3 Citations (Scopus)


Michor and Mumford showed that the mean curvature flow is a gradient flow on a Riemannian structure with a degenerate geodesic distance. It is also known to destroy the uniform density of gridpoints on the evolving surfaces. We introduce a related geometric flow which is free of these drawbacks. Our flow can be viewed as a formal gradient flow on a certain submanifold of theWasserstein space of probability measures endowed with Otto’s Riemannian structure. We obtain a number of analytic results concerning well-posedness and long-time stability which are, however, restricted to the 1D case of evolution of loops.

Original languageEnglish
JournalJournal of Geometric Analysis
Issue number4
Publication statusPublished - 2019
Externally publishedYes


  • Evolving surface
  • Gradient flow
  • Infinite-dimensional Riemannian manifold
  • Optimal transport
  • Volume

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