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Evolution equations like mean curvature flow, geometric heat flow and Willmore flow have been studied in our working group since quite a long time.
In doing so we have mainly focused on the finite element approximation.
Recently another example for an evolution equation on a (not necessarily embedded) Riemannian manifold has attracted our attention: Ricci flow
describes the evolution of a time-dependent Riemannian metric g(t) under the nonlinear partial differential equation
where ric(t) is the Ricci curvature with respect to g(t). Ricci flow is not parabolic itself, but it can be modified by the DeTurck trick into a
nonlinaer parabolic equation. Short time existence of Ricci flow was proved by Richard Hamilton, who introduced Ricci flow in 1982 in order
to gain insight into the topological classification of three dimensional smooth manifolds. Actually it plays an important role in
Grigori Perelman's proof of William Thurston's geometrization conjecture and the Poincaré conjecture respectively.
In contrast to its outstanding relevance in pure mathematics Ricci flow has not attracted appropriate attention in computational mathematics yet.
Previous results are restricted either to conformal geometry or to spherically symmetric metrics.
Therefore we plan to develop a numerical algorithm for the Ricci flow which doesn't make use of special assumptions like symmetries
and which principally works for higher dimensions though the main applications will be the two and three dimensional case. The basis for our numerical
methods will be an adequate discretization of Ricci flow. For this purpose we will discretize a weak form of the Ricci curvature by piecewise linear
finite elements. Here we profit from our experiences in the development of discretization techniques for geometric partial differential equations.
Particularly we can make use of the advantages of the finite element toolbox ALBERTA. A further goal of this project will be the improvement of present
visualization techniques
A concise introduction to Ricci flow is given in the book:
Lectures on the Ricci flow by Peter Topping, Cambridge University Press.
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