Differential Geometry Seminar: Locally volume collapsed 4-manifolds with respect to a lower sectional curvature bound
Seminar | October 30 | 1-2 p.m. | 891 Evans Hall
Thunwa Theerakarn, UC Berkeley
Perelman stated without proof that a 3-dimensional compact Riemannian manifold which is locally collapsed, with respect to a lower curvature bound, is a graph manifold. The statement was used to complete his proof of the geometrization conjecture. Kleiner-Lott gave a proof of this theorem as a part of their presentation of Perelman's proof of the geometrization conjecture.
In this talk, I will present a generalization of Perelman's local collapsing theorem to 4-dimensional Riemannian manifolds. Namely, a 4-dimensional closed orientable Riemannian manifold which is locally volume collapsed, with respect to a lower sectional curvature bound, admits a metric of nonnegative sectional curvature or an \(F\)-structure (under some regularity assumptions).