Abstract
In this paper we prove that every closed polyhedral surface in Euclidean three-space can be approximated (uniformly with respect to the Hausdorff metric) by smooth surfaces of the same topological type such that not only the (Gaussian) curvature but also the absolute curvature and the absolute mean curvature converge in the measure sense. This gives a direct connection between the concepts of total absolute curvature for both smooth and polyhedral surfaces which have been worked out by several authors, particularly N. H. Kuiper and T. F. Banchoff.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Banchoff, T. F.: ‘Critical Points and Curvature for Embedded Polyhedra’, J. Diff. Geom. 1 (1967), 257–268.
Banchoff, T. F.: ‘Critical Points and Curvature for Embedded Polyhedral Surfaces,’ Amer. Math. Monthly 77 (1970), 475–485.
Brehm, U. and Kühnel, W.: ‘Smooth Approximation of Polyhedral Surfaces with Respect to Curvature Measured,’ in Proc. Conf. Global Diff. Geom. and Global Analysis Berlin 1979, 64–68, Springer, 1981 (Lecture Notes in Mathematics Vol. 838).
Gritzmann, P.: ‘Tight Polyhedral Realisations of Closed 2-dimensional Manifolds in ℝ’, J. Geom. 17 (1981), 69–76.
Kühnel, W.: ‘Total Absolute Curvature of Polyhedral Manifolds with Boundary in E n’, Geom. Dedicata 8 (1979), 1–12.
Kuiper, N. H.: ‘Minimal Total Absolute Curvature for Immersions’, Inv. Math. 10 (1970), 209–238.
Kuiper, N. H.: ‘Morse Relations for Curvature and Tightness’, in Proc. Liverpool Sing. Symp. II, 77–89, Springer, 1971, (Lect. Notes in Math. Vol. 209).
Kuiper, N. H.: ‘Tight Topological Embeddings of the Moebiusband, J. Diff. Geom. 6 (1972), 271–283.
Langevin, R.: ‘Courbure et singularités Complexes’, Comm. Math. Helv. 54 (1979), 6–16.
Morse, M.: ‘Topologically Nondegenerate Functions’, Fund. Math. 88 (1975), 17–52.
Rourke, C. P. and Sanderson, B. J.: Introduction to Piecewise-Linear Topology, Springer, Berlin, Heidelberg, New York, 1972.
Schneider, R.: ‘Kritische Punkte und Krümmung für die Mengen des Konvexrings’, L' Enseignement Math. 23 (1977), 1–6.
Stone, D. A.: ‘Sectional Curvature in Piecewise Linear Manifolds’, Bull. Am. Math. Soc. 79 (1973), 1060–1063.
Stone, D. A.: ‘Geodesics in Piecewise Linear Manifolds’, Trans. Am. Math. Soc. 215 (1976), 1–44.
White, J. H.: ‘Minimal Total Absolute Curvature for Orientable Surfaces with Boundary’, Bull. Am. Math. Soc. 80 (1974), 361–362.
Author information
Authors and Affiliations
Additional information
The present paper is a detailed version of the short announcement [3].
Rights and permissions
About this article
Cite this article
Brehm, U., Kühnel, W. Smooth approximation of polyhedral surfaces regarding curvatures. Geom Dedicata 12, 435–461 (1982). https://doi.org/10.1007/BF00147585
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00147585