1/4-Pinched Contact Sphere Theorem

TL;DR

This paper proves that a closed contact 3-manifold with a 1/4-pinched sectional curvature must have a universally tight contact structure, improving previous results that required a 4/9-pinching constant.

Contribution

It establishes a sharper curvature pinching condition (1/4 instead of 4/9) for tightness of contact structures on 3-manifolds, advancing contact geometry understanding.

Findings

Contact structure is universally tight under 1/4-pinched curvature.

Improves the curvature pinching constant from 4/9 to 1/4.

Discusses tightness in positively curved contact open 3-manifolds.

Abstract

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.