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Research Report MRR97-004

The Minding Formula and its Application

Yi Fang

Abstract: We apply the Minding Formula for geodesic curvature and the Gauss-Bonnet Formula to calculate the total Gaussian curvature of certain 2-dimensional open complete (may be branched) Riemannian manifolds, the $\Cal M$ surfaces. We prove that the total curvature of an $\Cal M$ surface depends only on the Euler characteristic and the local behaviour of the metric at ends and branch points. Then we check that many important surfaces, such as complete minimal surfaces in $\R^n$ with finite total curvature, complete constant mean curvature surfaces in hyperbolic 3-space $H^3(-1)$ with finite total curvature, are actually branch point free $\Cal M$ surfaces. Therefore as corollaries we give simple proofs of some classical theorems such as the Chern-Osserman theorem for complete minimal surfaces in $\R^n$ with finite total curvature. For the reader's convenience, we also derive the Minding Formula.

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