Applications of geometric evolution equations.

Hi everybody,

I'm looking for applications of geometric evolution equations such as the Ricci flow and the extrinsic flows by Gauss and mean curvature. Applications other than topological applications such as geometrization are what I'm looking for and, even better if the application is to a non-mathematical area, such as Mullins' derivation of mean curvature flow as a model for the motion of `grain boundaries'.

Thanks,

ML

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I assume you also want to rule out theoretical physics? Else the entire field of general relativity is an "application of geometric evolution equations", and you also have things related to the renomalisation group flow. – Willie Wong Jul 8 2011 at 11:58
Agree with @Willie -- voting to close until community wikified. – Igor Rivin Jul 8 2011 at 14:39

Ricci flow has been applied to computer vision (e.g., to 3D shape matching), and to computer graphics (e.g., surface parametrization):

Wei Zeng, Samaras, D., Gu, D. , "Ricci Flow for 3D Shape Analysis," IEEE Transactions on Pattern Analysis and Machine Intelligence, 32(4) 2010 662-677.

Yang, Guo, Luo, Hu, Gu, "Generalized Discrete Ricci Flow," Computer Graphics Forum, Volume 28, Issue 7, pages 2005–2014, October 2009.

If you classify the eikonal equation as a geometric evolution equation, then there are many applications to the computer vision problem known as "shape from shading": deriving the 3D shape of an object from shading variation. For example:

Anna R. Bruss, "The eikonal equation: Some results applicable to computer vision," J. Math. Phys. 23, 890 (1982).

Ron Kimmel , James A. Sethian, "Optimal Algorithm for Shape from Shading and Path Planning," Journal of Mathematical Imaging and Vision,Volume 14 Issue 3, May 2001.

And in fact Sethian's book,

Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, 1996.

is filled with applications of the eikonal equation (and its variants): "The text includes applications from physics, chemistry, fluid mechanics, combustion, image processing, material science, seismology, fabrication of microelectronic components, computer vision and control theory."

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 Most interesting. Thanks Joseph! – kangdon Jul 8 2011 at 12:56

Many applications have been found for geometric evolution equations.

Yamabe flow was used in shape analysis which is related to medical research. See: "Detection of shape deformities using Yamabe flow and Beltrami coefficients" by Lok Ming Lui at al.

Inverse mean curvature flow was used by Huisken and Ilmanen to prove the Riemannian Penrose inequality. See: http://en.wikipedia.org/wiki/Riemannian_Penrose_inequality

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Two points regarding the Penrose Inequality: (a) Bray's proof also uses a (different) geometric flow (b) I consider the Riemannian Penrose Inequality (as well as the positive mass theorem) to be statements in pure mathematics (in)conveniently disguised as statements in physics.... :) – Willie Wong Jul 9 2011 at 10:39

The Willmore flow and the more complicated Helfrich flow are used to model cell membranes.

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