Immortal solution of the Ricci flow (Q2460727)

The authors consider the Ricci flow on a complex complete noncompact Kähler manifold \(M\) with nonnegative and bounded holomorphic bisectional curvature. A smooth solution of this flow is called immortal it it is defined for all \(t \geq 0\) and \(0 \leq R(x,t) \leq \frac{C}{1+t}\) where \(R(x,t)\) is the scalar curvature and \(C\) is some positive constant. In this article it is shown that an immortal solution exists if and only if \[ \int^r_0 sk(x,s)\,ds \leq C \log (2+r) \] for some constant \(C > 0\) and for all \(x \in M\) and \( r \geq 0\). Here \[ k(x,s) = \frac{1}{V(x,s)}\int_{B(x,s)} R(x) \,dx \] where \(B(x,s)\) is the geodesic ball of radius \(r\) with centre at \(x\), \(V(x,s)\) is the volume of this ball and \(R(x)\) is the scalar curvature of \(M\) with respect to the initial metric.