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The paper provides a different proof of the result of Brendle-Schoen on thedifferential sphere theorem. It is shown directly that the invariant cone ofcurvature operators with positive (or non-negative) complex sectional curvatureis 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 constantpositive curvature. Using earlier work of Yau and Zheng it can be shown that ametric with strictly (pointwise) 1/4-pinched sectional curvature has positivecomplex sectional curvature. This gives a direct proof of Brendle-Schoen'srecent differential sphere theorem, bypassing any discussion of positiveisotropic curvature.

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