Evolution equations of curvature tensors along the hyperbolic geometric flow

Abstract

The author considers the hyperbolic geometric flow $\tfrac{{\partial ^2 }}{{\partial t^2 }}g(t) = - 2Ric_{g(t)}$ introduced by Kong and Liu. Using the techniques and ideas to deal with the evolution equations along the Ricci flow by Brendle, the author derives the global forms of evolution equations for Levi-Civita connection and curvature tensors under the hyperbolic geometric flow. In addition, similarly to the Ricci flow case, it is shown that any solution to the hyperbolic geometric flow that develops a singularity in finite time has unbounded Ricci curvature.

Keywords

Hyperbolic geometric flow / Evolution equations / Singularity

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