Asian Journal of Mathematics

Volume 20 (2016)

Number 5

$1/4$-pinched contact sphere theorem

Pages: 893 – 902



Jian Ge (Beijing International Center for Mathematical Research, Beijing University, Beijing, China)

Yang Huang (Centre for Quantum Geometry of Moduli Spaces, Aarhus University, Aarhus, Denmark)


Given a closed contact $3$-manifold with a compatible Riemannian metric, we show that if the sectional curvature is $1/4$-pinched, then the contact structure is universally tight. This result improves the Contact Sphere Theorem in [EKM12], where a $4/9$-pinching constant was imposed. Some tightness results on positively curved contact open $3$-manifold are also discussed.


contact sphere theorem, curvature, tight contact structure

2010 Mathematics Subject Classification

Primary 53D10. Secondary 53B20.

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