Ricci flow preserves positive curvature on homogeneous spheres

Abstract: Since Hamilton introduced the Ricci flow in 1982, it has been a fundamental question to understand which curvature conditions are preserved under the flow. In this talk we focus on the notion of positive sectional curvature, denoted sec>0. Hamilton proved that sec>0 is preserved in dimensions 2 and 3. In contrast, there are examples of manifolds in dimensions 4, 6, 7, 12, 13 and 24 where sec>0 is not preserved. In the search of new examples it is natural to look at homogeneous manifolds. In this work we prove that spheres do not provide new examples, i.e. the flow preserves sec>0 on homogeneous spheres. We also show that the same holds for all compact rank one symmetric spaces. Together with previous works, this classifies simply connected manifolds which admit a homogeneous metric for which sec>0 is not preserved under the flow. This is joint work with Jason DeVito and Masoumeh Zarei.

differential geometrygeometric topologymetric geometry

Audience: researchers in the topic

Virtual seminar on geometry with symmetries

Series comments: Description: Research seminar in Lie group actions in Differential geometry.

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