A compactness property for solutions of the Ricci flow. (English) Zbl 0840.53029

The author considers families of complete Riemannian manifolds \((M,G)\) which are marked by fixing a point \(Q \in M\) and an orthonormal basis \(q\) at \(Q\). It is required that the injectivity radius at \(Q\) is uniformly bounded from below by some \(\rho > 0\) and that the curvature is uniformly bounded in norm. He shows that such a family of marked Riemannian manifolds is compact in the topology of \(C^\infty\)-convergence on compact sets (via identification by suitable diffeomorphisms) provided that all the covariant derivatives of the curvature are uniformly bounded as well. This extends and supplements earlier convergence results of Gromov, Peters, Green-Wu.
For solutions of the Ricci flow equation, bounds on the curvature automatically produce at subsequent times bounds on all the covariant derivatives of the curvature: This fact (due to Shi) is used to show compactness for families of solutions of the Ricci flow equation on a fixed interval \((A,\Omega)\) containing 0 with respect to the topology of smooth convergence on compact sets respecting a marking, provided that the curvature of the evolving Riemannian manifolds is uniformly bounded for all times \(t \in (A, \Omega)\) and that the injectivity radius at the markings is bounded from below by a positive number at the time \(t = 0\).


53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
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