Complete surfaces of at most quadratic area growth

P. Li
1997 Commentarii Mathematici Helvetici  
In this article, we study complete surfaces with finite topological type and has at most quadratic area growth. In particular, we show that if the curvature of such a surface does not change sign, then it must be of finite total curvature. Mathematics Subject Classification (1991). 53C20 Keywords. Riemannian manifold, Gaussian curvature, total curvature, finite topological type. In 1935, Cohn-Vossen [CV] studied the validity of the Gauss-Bonnet theorem for complete non-compact surfaces. In
more » ... t surfaces. In particular, he considered the relationship between the Euler characteristic χ(M ) and the integral of the Gaussian curvature K of a complete surface, M 2 , without boundary. He proved that if for any compact exhaustion {Ω i } of M , the limit M K = lim i→∞ Ωi K exists, then M K ≤ 2πχ(M ). Later on, in 1957, Huber [Hu1] proved that if the negative part of the Gaussian curvature of M defined by and M is conformally equivalent to a compact Riemann surface with finitely many punctures. In particular, this implies that M has finite topological type. It also implies that the positive part of the Gaussian curvature K + = max{0, K} is integrable and hence M must have finite total curvature, i.e.,
doi:10.1007/pl00000367 fatcat:ulv7ksiktzfwngeyz7p5o6fbrq