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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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