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Ricci deformation of the metric on complete non-compact Riemannian manifolds. (English) Zbl 0686.53037
Let (M,g) be an n-dimensional Riemannian manifold. Let $$R_{ijk\ell}=W_{ijk\ell}-V_{ijk\ell}+U_{ijk\ell}$$ be the decomposition of the Riemannian curvature tensor into the orthogonal components, where $$W_{ijk\ell}$$ is the Weyl conformal curvature tensor, and $$V_{ijk\ell}$$ and $$U_{ijk\ell}$$ denote the traceless Ricci part and the scalar curvature part respectively. Consider the evolution equation $(*)\quad (\partial /\partial t)g_{ij}(t)=- zR_{ij}(t),\quad G_{ij}(0)=g_{ij},$ where $$R_{ij}=R_{ikjk}$$ is the Ricci curvature tensor.
Suppose (M,g) is complete noncompact with $$n\geq 3$$. Assume that for any $$c_ 1,c_ 2>0$$ and $$\delta >0$$, there exists a constant $$\epsilon =\epsilon (n,c_ 1,c_ 2,\delta)>0$$ such that if the curvature of (M,g) satisfies: $$(A)\quad Vol(B(x_ 1,\gamma_ 1)\geq c_ 1\gamma^ n,\quad for\quad all\quad x\in M,\quad \gamma \geq 0\quad and\quad (B)| W_{ijk\ell}|^ 2+| V_{ijk\ell}|^ 2\leq \epsilon R^ 2,\quad 0<R\leq C_ 2/\gamma (x_ 0,x)^{2+\delta}\quad for\quad all\quad x\in M.$$Here we fixed a point $$x_ 0\in M$$ and for any $$x\in M$$, $$\gamma (x_ 0,x)$$ is the distance beteween $$x_ 0$$ and x. We set $$B(x,\gamma)=\{y\in M |$$ $$\gamma (x,y)<\gamma \}$$. Then it is proved, that the evolution equation (*) has a solution for all time $$0\leq t<+\infty$$ and the metric $$g_{ij}(t)$$ converges to a smooth metric $$g_{ij}(\infty)$$ as time $$t\to +\infty$$ such that $$R_{ijk\ell}(\infty)\equiv 0$$ on M.
Reviewer: T.Ochiai

##### MSC:
 53C20 Global Riemannian geometry, including pinching 58J35 Heat and other parabolic equation methods for PDEs on manifolds
##### Keywords:
evolution equation; Ricci curvature tensor
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