Diffeomorphisms of the circle and hyperbolic curvature (Q2761197)

Let \(Tf\) be the hyperbolic trace of a smooth function \(f\) of a real or complex variable. Let \(\gamma(t)\) be a convex curve of class \(C^2\) in the unit disc with the Poincaré metric. The author shows that \(Tf\) is a conjugacy invariant under Möbius transformations. Moreover, he associates with the convex curve \(\gamma(t)\) a diffeomophism \(f\) of the circle and proves that the trace of the diffeomorphism \(T(f)\) is twice the reciprocal of the geodesic curvature of \(\gamma(t)\).NEWLINENEWLINEIt is known from the classical four-vertex theorem (proved by Mukhopadhyaya in 1909) that a simple closed curve of class \(C^2\) with nonvanishing curvature has at least four extrema of the curvature. Over the years, the theorem has been generalized by many authors. In the present paper, the author uses the theorem of Ghys on Schwarzian derivatives to give an entirely new and very elegant proof of the four-vertex theorem for closed convex curves in the hyperbolic plane.