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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

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