Applications of geometric evolution equations. - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T04:22:55Z http://mathoverflow.net/feeds/question/69769 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69769/applications-of-geometric-evolution-equations Applications of geometric evolution equations. kangdon 2011-07-08T05:10:13Z 2011-07-09T04:13:17Z <p>Hi everybody,</p> <p>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'.</p> <p>Thanks,</p> <p>ML</p> http://mathoverflow.net/questions/69769/applications-of-geometric-evolution-equations/69784#69784 Answer by Joseph O'Rourke for Applications of geometric evolution equations. Joseph O'Rourke 2011-07-08T11:10:12Z 2011-07-08T13:03:12Z <p>Ricci flow has been applied to computer vision (e.g., to 3D shape matching), and to computer graphics (e.g., surface parametrization):</p> <blockquote> <p>Wei Zeng, Samaras, D., Gu, D. , "Ricci Flow for 3D Shape Analysis," <a href="http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=5374410" rel="nofollow"><em>IEEE Transactions on Pattern Analysis and Machine Intelligence</em>, 32(4) 2010 662-677</a>.</p> <p>Yang, Guo, Luo, Hu, Gu, "Generalized Discrete Ricci Flow," <a href="http://onlinelibrary.wiley.com/doi/10.1111/j.1467-8659.2009.01579.x/abstract" rel="nofollow"><em>Computer Graphics Forum</em>, Volume 28, Issue 7, pages 2005–2014, October 2009</a>.</p> </blockquote> <p>If you classify the <a href="http://en.wikipedia.org/wiki/Eikonal_equation" rel="nofollow">eikonal equation</a> 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:</p> <blockquote> <p>Anna R. Bruss, "The eikonal equation: Some results applicable to computer vision," <a href="http://jmp.aip.org/resource/1/jmapaq/v23/i5/p890_s1?isAuthorized=no" rel="nofollow"><em>J. Math. Phys.</em> 23, 890 (1982)</a>.</p> <p>Ron Kimmel , James A. Sethian, "Optimal Algorithm for Shape from Shading and Path Planning," <a href="http://portal.acm.org/citation.cfm?id=570166" rel="nofollow"><em>Journal of Mathematical Imaging and Vision</em>,Volume 14 Issue 3, May 2001</a>.</p> </blockquote> <p>And in fact Sethian's book, </p> <blockquote> <p><a href="http://math.berkeley.edu/~sethian/2006/Publications/Book/1996/book_1996.html" rel="nofollow"><em>Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science</em></a>, Cambridge University Press, 1996.</p> </blockquote> <p>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."</p> http://mathoverflow.net/questions/69769/applications-of-geometric-evolution-equations/69836#69836 Answer by Paul for Applications of geometric evolution equations. Paul 2011-07-08T22:51:29Z 2011-07-08T22:51:29Z <p>Many applications have been found for geometric evolution equations. </p> <p>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. </p> <p>Inverse mean curvature flow was used by Huisken and Ilmanen to prove the Riemannian Penrose inequality. See: <a href="http://en.wikipedia.org/wiki/Riemannian_Penrose_inequality" rel="nofollow">http://en.wikipedia.org/wiki/Riemannian_Penrose_inequality</a></p> http://mathoverflow.net/questions/69769/applications-of-geometric-evolution-equations/69848#69848 Answer by timur for Applications of geometric evolution equations. timur 2011-07-09T04:13:17Z 2011-07-09T04:13:17Z <p>The <a href="http://en.wikipedia.org/wiki/Willmore_energy" rel="nofollow">Willmore flow</a> and the more complicated Helfrich flow are used to model cell membranes.</p>