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The Journal of Geometric Analysis

, Volume 29, Issue 4, pp 3055–3097 | Cite as

Uniformly Compressing Mean Curvature Flow

  • Wenhui Shi
  • Dmitry VorotnikovEmail author
Article

Abstract

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 the Wasserstein 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.

Keywords

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

Mathematics Subject Classification

35A01 35A02 53C44 58E99 

Notes

Acknowledgements

The first author would like to thank Herbert Koch for helpful discussions related to local and global well-posedness issues. The research was partially supported by CMUC— UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MCTES and by the ERDF through PT2020, and by the FCT (TUBITAK/0005/2014).

Compliance with Ethical Standards

Conflict of interest

We have no conflict of interest to declare.

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Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.CMUC, Department of MathematicsUniversity of CoimbraCoimbraPortugal

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