Geometric Analysis Seminar: Herng Yi Cheng (Toronto)
- Event Type
- Seminar/Symposium
- Sponsor
- Department of Mathematics
- Location
- Altgeld 241
- Virtual
- Join online
- Date
- Nov 6, 2025 3:00 pm
- Speaker
- Herng Yi Cheng (Toronto)
- Contact
- Pierre Albin, Eric Chen, Pei-Ken Hung, Gabriele La Nave
- palbin@illinois.edu, ecchen@illinois.edu, pkhung@illinois.edu, lanave@illinois.edu
- Views
- 24
- Originating Calendar
- Geometric Analysis Seminars
Title: Stable 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 R^{n+1} contain a stable closed geodesic, i.e. a closed geodesic with Morse index zero? This is impossible for even n by a classical theorem of Synge, but I will construct such M with stable closed geodesics for all odd n≥3.
I will also construct closed convex hypersurfaces M of R^{n+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)