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International Mathematics Research Notices Advance Access published online on September 12, 2009
International Mathematics Research Notices, doi:10.1093/imrn/rnp147
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© The Author 2009. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

Isotropic Curvature and the Ricci Flow

Huy T. Nguyen

Centre for Mathematics and its Applications, Australian National University, ACT 0200 Australia and Max Planck Institute for Gravitational Physics (Albert Einstein Institute), am Mühlenberg 1, D-14476 Golm, Germany

Correspondence: Correspondence to be sent to: huy.nguyen{at}aei.mpg.de

In this paper, we study the Ricci flow on higher dimensional compact manifolds. We prove that nonnegative isotropic curvature is preserved by the Ricci flow in dimensions greater than or equal to four. In order to do so, we introduce a new technique to prove that curvature functions defined on the orthonormal frame bundle are preserved by the Ricci flow. At a minimum of such a function, we compute the first and second derivatives in the frame bundle. Using an algebraic construction, we can use these expressions to show that the nonlinearity is positive at a minimum. Finally, using the maximum principle, we can show that the Ricci flow preserves the cone of curvature operators with nonnegative isotropic curvature.

Communicated by John Lott


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