Title: On Yau’s pinching problem

 

 

Abstract: In 1990, Yau asked in his “Open problems in geometry": “The famous pinching problem says that on a compact simply connected manifold if \(K_\min > 1/4 K_\max\) (i.e., the minimum of the sectional curvature is bigger than one quarter of its maximum), then the manifold is homeomorphic to a sphere. If we replace \(K_\max\) by normalized scalar curvature (equal to the average of sectional curvatures), can we deduce similar pinching theorems?” In this talk, we present a new partial result to Yau’s pinching problem that improves earlier bounds of Gu–Xu, both by sharpening the pinching constants and by weakening the curvature quantities involved. The key observation is a new link between sectional curvature pinching and positive isotropic curvature.