Ricci deformation of the metric on complete non-compact Riemannian manifolds
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.
scientific article; zbMATH DE number 4124528