Riemannian 3-spheres that are hard to sweep out by short curves

YMSC-BJTU Geometry seminar

Organizers：

Han Hong, Zhifei Zhu

Speaker：

Omar Alshawa (University of Toronto)

Time：

Wed., 10:00-11:00 am, Mar. 12, 2025

Online：

Zoom meeting ID: 890 9835 3295

Password: 111111

Title：

Riemannian 3-spheres that are hard to sweep out by short curves

Abstract：

Does every Riemannian 3-sphere M contain a closed geodesic whose length is bounded from above by some function f(d,V) of the diameter d and volume V of M? One strategy to find such a closed geodesic is to construct a sweepout of M by closed curves of length at most f(d,V). In collaboration with Herng Yi Cheng, we prove that this method of finding short closed geodesics does not work for a certain class of sweepouts.

Let L>0 be large. We show that there exists M of diameter and volume less than 1 such that for any sweepout of M by closed curves within this class, one of the curves must be longer than 1.