Project B2: Geometric evolution equations



Principal investigators

Participating scientists

Summary

The topic of this project is to understand Ricci flow in higher dimensions.
The Ricci flow is a geometric (weakly parabolic) evolution equation on the space of Riemannian metrics of a smooth compact manifold, which tends to improve the geometry. This flow has been introduced by Hamilton to prove that a compact 3-dimensional Riemannian manifold which admits a Riemannian metric with positive Ricci curvature is diffeomorphic to a 3-dimensional spherical space form.

In general Ricci flow will develop finite time singularities, which have resisted being understood for over twenty years. In dimension 3, by the revolutionary work of Perelman in 2002 and 2003, these singularities could be classified. Based on this, Ricci flow with surgery has been introduced by Perelman to prove Thurstan's geometrization conjecture and in particular the Poincaré conjecture.
It should be pointed out that in dimension 3 Ricci flow with surgery can be defined for any initial metric on a compact manifold. Currently it seems to be hopeless to generalize this to higher dimensions. For instance the corresponding elliptic problem - classification of Einstein manifolds/Ricci solitons - is completely open in dimensions 5 and higher. In higher dimensions it is therefore manifest only to consider initial metrics which satisfy certain "natural" curvature conditions, similarly to Hamilton's original approach in dimension 3.
On a first level this has already been achieved: By now several curvature conditions are known which force the (normalized) Ricci flow to converge to a round metric for initial metrics satisfying such curvature conditions. One important result in this direction is the proof of the differentiable sphere theorem by Brendle and Schoen, a longstanding conjecture in differential geometry.
On a second level one is aiming for Ricci flow invariant curvature conditions, which do not lead to convergence results but allow Ricci flow to develop singularities. These invariant curvature conditions should however be "small" enough to guarantee that the singularity analysis of the Ricci flow can be carried out.
The project is subdivided into the following subprojects:
  • Second variation and a connection to Lie algebras
  • New invariant curvature condition
  • Weak Lyapunov functions
  • Singularity analysis
  • Miscellaneous

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