A note on the Gaussian curvature on noncompact surfaces

2016/09/24 by S. Cecchini, Cecchini, Simone
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1609.07631

openalex publication_date 2016/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a short proof of the following fact. Let Σ be a connected, finitely connected, noncompact manifold without boundary. If g is a complete Riemannian metric on Σ whose Gaussian curvature K is nonnegative at infinity, then K must be integrable. In particular, we obtain a new short proof of the fact that if Σ admits a complete metric whose Gaussian curvature is nonnegative and positive at one point, then Σ is diffeomorphic to ℝ².

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