A note on Kähler-Ricci flow
The author considers a complete Kähler-Ricci flow \((M,g(t))\) satisfying Shi's estimate \[ \| \nabla^k Rm \|^2 (x,t) \leq \frac{C(k,s)}{t^k}; \qquad \quad k \in \mathbb N, s \in (0,T), (x,t) \in M \times (0,s], \] where \(T\) is the maximal existence time. Assuming uniform equivalence of \(g(t)\) to \(g(0)\) in the sense that \[ C^{-1} g(0) \leq g(t) \leq C g(0) \] for some given \(C > 0\), the author proves that for any \(k \in \mathbb N\) there exist constants \(A_, B_k\) such that \[ \| \nabla^k Rm \|^2 (x,t) \leq A_k + \frac{B_k}{t^k}, \qquad (x,t) \in M \times [0,T). \] The proof relies on the maximum principle and an induction procedure. An application is given to the long-time existence problem.
scientific article; zbMATH DE number 6097139