Metric deformation of non-positively curved manifolds (Q917055)

Let (M,g) be a complete Riemannian manifold with semidefinite curvature \((K_ g\leq 0\) or \(K_ g\geq 0)\). In this paper conditions for the metric g to be (conformally) deformable to a metric \(\bar g\) of definite curvature \((K_{\bar g}<0\) or \(K_{\bar g}>0)\) are established. For \(K_ g\leq 0\) (resp. \(K_ g\geq 0)\) the results obtained are as follows: Let \(p\in M\). Then there exists a positive number R which is determined by g and its derivatives around p such that the following holds: if \(K_ g<0\) (resp. \(K_ g>0)\) on \(B_ R(p)\) then there is a metric \(\bar g\) such that \(K_{\bar g}<0\) (resp. \(K_{\bar g}>0)\) and \(g=\bar g\) on \(M- B_ R(p)\), where \(B_ R(p)=\{q\in M\); \(d(p,q)<R\}\). The author also conjectures that if g is a complete metric with \(K_ g\leq 0\) on \(R^ n\) (n\(\geq 3)\) and if a compact A with \(K_ g<0\) on \(R^ n-A\) exists, then a complete metric \(\bar g\) exists with \(K_{\bar g}<0\) and \(g=\bar g\) on \(R^ n-B\) for some compact B.