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Mean curvature flow


Authors: Tobias Holck Colding, William P. Minicozzi II and Erik Kjær Pedersen
Journal: Bull. Amer. Math. Soc. 52 (2015), 297-333
MSC (2010): Primary 53C44
DOI: https://doi.org/10.1090/S0273-0979-2015-01468-0
Published electronically: January 13, 2015
MathSciNet review: 3312634
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Abstract: Mean curvature flow is the negative gradient flow of volume, so any hypersurface flows through hypersurfaces in the direction of steepest descent for volume and eventually becomes extinct in finite time. Before it becomes extinct, topological changes can occur as it goes through singularities. If the hypersurface is in general or generic position, then we explain what singularities can occur under the flow, what the flow looks like near these singularities, and what this implies for the structure of the singular set. At the end, we will briefly discuss how one may be able to use the flow in low-dimensional topology.


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Additional Information

Tobias Holck Colding
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachu- setts Avenue, Cambridge, Mssachusetts 02139-4307
Email: colding@math.mit.edu

William P. Minicozzi II
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachu- setts Avenue, Cambridge, Mssachusetts 02139-4307
Email: minicozz@math.mit.edu

Erik Kjær Pedersen
Affiliation: Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
Email: erik@math.ku.dk

DOI: https://doi.org/10.1090/S0273-0979-2015-01468-0
Received by editor(s): August 27, 2012
Received by editor(s) in revised form: June 11, 2014
Published electronically: January 13, 2015
Additional Notes: The first two authors were partially supported by NSF Grants DMS 11040934, DMS 0906233, and NSF FRG grants DMS 0854774 and DMS 0853501
Article copyright: © Copyright 2015 American Mathematical Society

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