Mathematics – Differential Geometry
Scientific paper
2011-07-04
Mathematics
Differential Geometry
42 pages, Added: a) example of an immortal solution which flows from bounded to unbounded curvature in finite time, b) long ti
Scientific paper
We prove short time existence for the Ricci flow on open manifolds of nonnegative complex sectional curvature. We do not require 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 nonnegative complex sectional curvature which subconverge to a solution of the Ricci flow on the open manifold. Furthermore, we find an optimal volume growth condition which guarantees long time existence, and we give an analysis of the long time behaviour of the Ricci flow. Finally, we construct an explicit example of an immortal nonnegatively curved solution of the Ricci flow with unbounded curvature for all time.
Cabezas-Rivas Esther
Wilking Burkhard
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