Mass of rays on complete open surfaces

Abstract

The total curvature of a complete open surface describes certain properties of the Riemannian structure which defines it. We study relationships between the total curvature and the mass of rays on a finitely connected complete open surface and obtain some integral formulas. 0. Introduction. Throughout this paper let M be a connected, finitely connected, oriented, complete and noncompact Riemannian 2-manifold without boundary. The total curvature c(M) of M is defined to be an improper integral over M of Gaussian curvature G with respect to the area element dM of M. A well-known theorem due to Cohn-Vossen [1] states that if M admits total curvature, then 2πχ(M) — c(M) > 0, where χ(M) is the Euler characteristic of M. Clearly c(M) depends on the choice of Riemannian metric. This phenomenon gives rise to the idea that the value 2πχ(M) - c{M) should describe certain properties of Riemannian metric which defines it. A ray (respectively, a straight line) on M is by definition a unit speed geodesic parametrized on [0, oo) (respectively, on R) every subarc of which realizes distance between its terminal points. For a point p G M let Sp(l) be the unit circle centered at the origin of the tangent space Mp to M at p. Let A(p) be the set of all unit vectors tangent to rays emanating from p. A(p) is closed in Sp(l). Let ΐDl be the natural measure on Sp(l) induced from the Riemannian metric. A relation between the mass of rays and the total curvature was first investigated by Maeda in [6], [7]. He proved that if M is homeomorphic to R2 and if G > 0, then ΐOloA > 2π — c(M), and in particular inf^ SDT ° A = 2π - c(M) . These results were extended by Shiga in [10], [11] to Riemannian planes whose Gaussian curvatures change sign, and later by Oguchi [9] to finitely connected M with one endpoint. In connection with an isoperimetric problem discussed by Fiala [3] and Hartman [4], the first-named author proved in [14] that if M has one end and if 2πχ(M) - c(M) < 2π , then for every monotone increasing sequence {Kj} of compact sets with \JKj = M,