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The gradient structure of the mean curvature flow
Published/Copyright:
March 15, 2014
Abstract
The concept of curve of maximal slope, a generalized notion of gradient flow, is extended to the setting of length spaces, which includes all metric spaces. This extended definition is used to show that curves of maximal slope in a metric space do not depend on the full metric, but only on the concept of curve length generated by it. Subsequently, it is shown that a length space can be constructed to describe L2 flow: geometric flows where the energy dissipated by a moving surface is the L2-norm of its normal velocity. Finally, it is shown that the mean curvature flow is the gradient flow of the area functional in this space.
Received: 2013-9-23
Revised: 2014-2-17
Accepted: 2014-2-24
Published Online: 2014-3-15
Published in Print: 2015-7-1
© 2015 by De Gruyter
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