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The Harnack estimate for the Ricci flow. (English) Zbl 0804.53023
The author proves the following theorem: Let $$g_{ij}$$ be a complete solution with bounded curvature to the Ricci flow $${\partial g_{ij}\over \partial t}= -2 R_{ij}$$ on a manifold $$M$$ in some time interval $$0< t< T$$ and suppose $$g_{ij}$$ has a weakly positive curvature operator, so that $$R_{ijkl} U_{ij} U_{kl}\geq 0$$ for every two-form $$U_{ij}$$. With $$P_{ijk}= D_ i R_{jk}- D_ j R_{ik}$$ and $$M_{ij}= \Delta R_{ik}- {1\over 2} D_ i D_ j R+ 2 R_{ikjl} R_{kl}- R_{ik} R_{jk}+{1\over 2t} R_{ij}$$. Then for all one-forms $$W_ i$$ and all two-forms $$U_{ij}$$ one has the generalized “Harnack inequality” $$M_{ij} W_ i W_ j+ 2 P_{ijk} U_{ij} W_ k+ R_{ijkl} U_{ij} U_{kl}\geq 0$$.

##### MSC:
 53C20 Global Riemannian geometry, including pinching 58J35 Heat and other parabolic equation methods for PDEs on manifolds 58E11 Critical metrics
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