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Deforming the metric on complete Riemannian manifolds. (English) Zbl 0676.53044
Let \((M^ n,g)\) be a complete Riemannian manifold with its Riemann curvature tensor R bounded by \(k_ 0\). The author proves that there exists a constant \(T=T(k_ 0,n)\) such that the evolution equation \[ (\partial /\partial t)g(x,t)=-2 Ric(x,t);\quad g(x,0)=g(x) \] (where Ric denotes the Ricci tensor) has a smooth solution \(g(x,t)>0\) for \(0\leq t\leq T\). Moreover, for any integer \(m\geq 0\) there is a constant \(c_ m=c_ m(k_ 0,n)\) such that the mth derivative of R is bounded by \(c_ mt^{-m}\) (0\(\leq t\leq T)\). It follows that for (M,g) as above there is a metric \(\tilde g\) on M equivalent to g (i.e. \(c^{-1}g\leq \tilde g\leq cg\) for some constant c) such that \(\tilde R\) has all derivatives w.r.t. \(\tilde g\) bounded. The case of \(| R|\) having polynomial growth is also discussed.
Reviewer: M.Lewkowicz

53C20 Global Riemannian geometry, including pinching
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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