Frontiers of Mathematics in China

, Volume 3, Issue 4, pp 475–494

Certain 4-manifolds with non-negative sectional curvature

Research Article

DOI: 10.1007/s11464-008-0037-6

Cite this article as:
Cao, J. Front. Math. China (2008) 3: 475. doi:10.1007/s11464-008-0037-6
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Abstract

In this paper, we study certain compact 4-manifolds with non-negative sectional curvature K. If s is the scalar curvature and W+ is the self-dual part of Weyl tensor, then it will be shown that there is no metric g on S2 × S2 with both (i) K > 0 and (ii) \( {\textstyle{1 \over 6}}s - W_ + \ge 0 \). We also investigate other aspects of 4-manifolds with non-negative sectional curvature. One of our results implies a theorem of Hamilton: “If a simply-connected, closed 4-manifold M4 admits a metric g of non-negative curvature operator, then M4 is one of S4, \( \mathbb{C}\rm P^2\) and S2 × S2”. Our method is different from Hamilton’s and is much simpler. A new version of the second variational formula for minimal surfaces in 4-manifolds is proved.

Keywords

Sectional curvature scalar curvature Weyl tensor minimal surface 4-manifold 

MSC

53C20 53C21 58C99 

Copyright information

© Higher Education Press and Springer-Verlag GmbH 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Notre DameNotre DameUSA

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