Geometric flows have many applications in physics and geometry.
The mean curvature flow occurs in the description of the interface
evolution in certain physical models. This is related to the
property that such a flow is the gradient flow of the area
functional and therefore appears naturally in problems where a
surface energy is minimized. The mean curvature flow also has many
geometric applications, in analogy with the Ricci flow of metrics
on abstract riemannian manifolds. One can use this flow as a tool
to obtain classification results for surfaces satisfying certain
curvature conditions, as well as to construct minimal surfaces.
Geometric flows, obtained from solutions of geometric parabolic
equations, can be considered as an alternative tool to prove
isoperimetric inequalities. On the other hand, isoperimetric
inequalities can help in treating several aspects of convergence of
these flows. Isoperimetric inequalities have many applications in
other fields of geometry, like hyperbolic manifolds.
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