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)