Positive Complex Sectional Curvature, Ricci Flow and the Differential Sphere Theorem

Details Positive Complex Sectional Curvature, Ricci Flow and the Differential Sphere Theorem Positive Complex Sectional Curvature, Ricci Flow and the Differential Sphere Theorem

2007-06-03

arxiv.org/abs/0706.0332v1

Mathematics

Differential Geometry

Scientific paper

The paper provides a different proof of the result of Brendle-Schoen on the differential sphere theorem. It is shown directly that the invariant cone of curvature operators with positive (or non-negative) complex sectional curvature is preserved by the Ricci flow. This implies, by a result of B\"ohm-Wilking, that the normalized Ricci flow deforms such a metric to a metric of constant positive curvature. Using earlier work of Yau and Zheng it can be shown that a metric with strictly (pointwise) 1/4-pinched sectional curvature has positive complex sectional curvature. This gives a direct proof of Brendle-Schoen's recent differential sphere theorem, bypassing any discussion of positive isotropic curvature.

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