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How to produce a Ricci flow via Cheeger–Gromoll exhaustion

Esther Cabezas-Rivas;  Burkhard Wilking — 2015

Journal of the European Mathematical Society

We prove short time existence for the Ricci flow on open manifolds of non-negative complex sectional curvature without requiring upper curvature bounds. By considering the doubling of convex sets contained in a Cheeger–Gromoll convex exhaustion and solving the singular initial value problem for the Ricci flow on these closed manifolds, we obtain a sequence of closed solutions of the Ricci flow with non-negative complex sectional curvature which subconverge to a Ricci flow on the open manifold. Furthermore,...

Non-negative curvature obstructions in cohomogeneity one and the Kervaire spheres

Karsten Grove;  Luigi Verdiani;  Burkhard Wilking;  Wolfgang Ziller — 2006

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In contrast to the homogeneous case, we show that there are compact cohomogeneity one manifolds that do not support invariant metrics of non-negative sectional curvature. In fact we exhibit infinite families of such manifolds including the exotic Kervaire spheres. Such examples exist for any codimension of the singular orbits except for the case when both are equal to two, where existence of non-negatively curved metrics is known.

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