Abstract.
In this paper, the negative gradient flow for the L 2-integral of Ricci curvature on a 3-manifold is considered. It is not known whether the solution to this fourth order geometric evolution equation exists, and whether it will develop singularities in finite time. Based on the trick of De Turck and the idea of Hamilton on the flow of Ricci curvature, the local existence on any compact Riemannian manifold is obtained. In addition, the conditions for the occurences of singularities in finite time during the evolution and the asymptotic behavior of the flow on a 3-manifold are discussed.
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Received: 22 April 2002 / Revised version: 5 January 2003 Published online: 24 April 2003
This work is partially supported by the Foundations of NNSF of China, the Foundation for University Key Teacher by MEC and Shanghai Priority Academic Discipline.
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Yu, Z. The negative gradient flow for the L 2-integral of Ricci curvature. manuscripta math. 111, 163–186 (2003). https://doi.org/10.1007/s00229-003-0363-7
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DOI: https://doi.org/10.1007/s00229-003-0363-7