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A geometric interpretation of Hamilton’s Harnack inequality for the Ricci flow. (English) Zbl 0856.53030
Let \((M,g)\) be a complete Riemannian manifold with bounded curvature, and let \((M,g(t))\) be a solution to the Ricci flow \((\partial/ \partial t) g_{ij}=-2R_{ij}\) with \(g(0) = g\). We define a symmetric 2-tensor \(M_{ij}\) and a 3-tensor \(P_{ijk}\) by \(M_{ij}=\Delta R_{ij}-(1/2) \nabla_i \nabla_jR+2R_{kijl} R_{kl}-R_{ik}R_{kj}\) and \(P_{ijk} = \nabla_iR_{jk} - \nabla_jR_{ik}\), where \(R_{ijkl}=g_{lm}R^m_{ijk}\).
Hamilton’s Harnack inequality says that if \((M,g(t))\) is a solution to the Ricci flow with semi-positive curvature operator, then for any 1-form \(W_i\) and 2-form \(U_{ij}\) we have \[ M_{ij} W_iW_j-2P_{ijk} U_{ij} W_k+R_{ijkl} U_{ij} U_{lk}\geq-{1\over 2t}R_{ij} W_iW_j. \] The authors try to give a geometric interpretation of this inequality. In fact, they show, among other things, that the left hand side of the above inequality, which is called the Harnack quantity, is the curvature of a torsion free connection compatible with a degenerate metric \(\widetilde g\) on the spacetime \(M \times [0,T)\), given for \(\tau \in[0,T)\) by \(\widetilde g_\tau(x,t) = g^{-1}(x,t+\tau)\) for \((x,t)\in M\times [0,T-\tau)\), where \(g^{-1}\) is the inverse of the metric \(g\) and where \([0,T)\) is the time interval of existence of the solution to the Ricci flow.

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