Herng Yi Cheng, University of Toronto

Title: Index-zero closed geodesics and stable geodesic nets in convex hypersurfaces

Abstract: Can a convex body be caught using a lasso? More formally, can a closed and convex hypersurface M of Rn+1 contain a closed geodesic with Morse index zero? This is impossible for even n by a theorem of Synge. I will construct such M with index-zero closed geodesics for all odd n ≥ 3.

I will also construct, for the first time, closed convex hypersurfaces M of Rn+1 of every dimension n ≥ 3 that contain "stable geodesic nets". These are embedded graphs whose images must lengthen when perturbed slightly. They can be thought of as nets of rope that "capture" convex bodies. The Lawson-Simons conjecture would imply that M cannot contain stable geodesic nets if its curvature is 1/4-pinched.

These constructions use a new method of building explicit billiard trajectories in convex polytopes with "twisted parallel transport." (arXiv:2109.09377, arXiv:2203.07166)