On a relation between the total curvature and the measure of rays (Q788295)

A well-known theorem of Cohn-Vossen says that for a non-compact complete surface M the following inequality holds \(\int K\leq 2\pi e(M)\), provided the total curvature \(\int K\) exists. Here K is the Gaussian curvature and e(M) is the Euler characteristic of M. In this paper the author proves that 2\(\pi\) e(M)-\(\int K\) is an upper bound for the measure of all initial unit vectors of all the geodesic rays (length minimising) emanating from a given point. He also proves that if M is homeomorphic to \({\mathbb{R}}^ 2\), then 2\(\pi\)-\(\int K_+\) is a lower bound for this measure, where \(K_+=\max(K,0)\). The method of proof is application of the Gauss-Bonnet formula to a well chosen sequence of compact domains exhausting M. The proof relies on classical ideas of Cohn-Vossen and Busemann.