Title: Tangent lines, inflections, and vertices of closed curves
Speaker: Mohammad Ghomi, Georgia Tech
Abstract: We show that every smooth closed curve C immersed in Euclidean 3-space satisfies the sharp inequality 2(P+I)+V >5 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of C. The proof, which employs curve shortening flow with surgery, is based on corresponding inequalities for the numbers of double points, singularities, and inflections of closed curves in the real projective plane which intersect every closed geodesic. These findings extend some classical results in curve theory including works of Moebius, Fenchel, and Segre, which is also known as Arnold's "tennis ball theorem”.
