A sharp characterization of the smooth 4-sphere in curvature terms

Abstract

This article presents a sharp pointwise geometric characterisation of the smooth structure of 5 4 and its only standard quotient EP 4 , which we obtained some years ago.The proof that we give here is a significantly simplified version of our original proof ([M2]).The main statements are the following two theorems.Theorem 1.Any compact 4-Tnanifold which admits a metric whose curvature is more pinched (w.r.t."weak pinching", see below) than the product S 1 x (iS 3 , can) is diffeomorphic to the (standard) 4-sphere, if it is orientable, or else to the standard Za-quotient'.A. Splitting of the fundamental polynomial.B. Decoupling Ricci and Weyl curvatures.C. Prom P -weak pinching to weak pinching.III.Solving the algebraic reduction (first steps): a sharp bound for the fundamental polynomial whose dependence on the Ricci curvature is scalar.A. Some symmetries of the fundamental polynomial.B. Dependence on the distorsion in the lower two Z 2 -eigenvalues.C. Dependence on the largest Z 2 eigenvalue.IV. Solving the algebraic reduction (continued) : isolating S 1 x S^-type curvatures.A. Eliminating some dependence on the Weyl spectrum.B. Some further elimination w.r.t. the Weyl spectrum and Ricci curvature.V. Solving the algebraic reduction (end) : isolating P 2 -type curvatures.A. Eliminating some dependence on the Weyl spectrum.B. Eliminating some more dependence : the case where the tracefree Ricci curvature is large.C. Eliminating some more dependence : the case where the tracefree Ricci curvature is small.VI. End of the proof.A. Behavior of the gradient of the scalar curvature.

Locations

• Communications in Analysis and Geometry - View - PDF