Positive sectional curvature on the exotic $8$-sphere and order-three homotopy $10$-spheres

Abstract

We prove that the exotic smooth $8$-sphere and both oriented homotopy $10$-spheres representing elements of order three admit Riemannian metrics with strictly positive sectional curvature. The construction combines Sperança's special $S^3$-$S^3$ bundle models with the compatible-disk construction of He, Liu and Yau. We show that the relevant bundles admit equivariant polar normal forms, with transition functions that are constant along meridians and conjugation-equivariant. We also show that the representation-dependent part of the He--Liu--Yau construction requires only a uniform bound on the infinitesimal action fields. Sperança's $8$- and $10$-dimensional examples satisfy this bound. The resulting northern and southern metrics have matching boundary metrics and compatible second fundamental forms, so the Reiser--Wraith gluing theorem gives the required positively curved metrics.