In this work, we obtain a local maximum principle along the Ricci flow $g(t)$ under the condition that $\mathrm {Ric}(g(t))\le {\alpha } t^{-1}$ for $t>0$ for some constant ${\alpha }>0$ . As an application, we will prove that under this condition, various kinds of curvatures will still be nonnegative for $t>0$ , provided they are non-negative initially. These extend the corresponding known results for Ricci flows on compact manifolds or on complete noncompact manifolds with bounded curvature. By combining the above maximum principle with the Dirichlet heat kernel estimates, we also give a more direct proof of Hochard’s [15] localized version of a maximum principle by Bamler et al. [1] on the lower bound of different kinds of curvatures along the Ricci flows for $t>0$ .

在这项工作中，我们在 $ \mathrm {Ric}(g(t))\le {\alpha } t^{-1}$ for $t>0$ 对于一些常数 ${\alpha }>0$ 。作为一个应用，我们将证明在这种情况下，各种曲率对于 $t>0$ 仍然是非负的，前提是它们最初是非负的。这些扩展了 Ricci 流在紧流形或具有有界曲率的完全非紧流形上的相应已知结果. 通过将上述最大原理与 Dirichlet 热核估计相结合，我们还更直接地证明了 Hochard [15] 由 Bamler 等人提出的最大原理的本地化版本。 [1] 关于$t>0$ 沿 Ricci 流的不同类型曲率的下界 。