Lipschitzian mappings and total mean curvature of polyhedral surfaces. I

Author:
Ralph Alexander

Journal:
Trans. Amer. Math. Soc. **288** (1985), 661-678

MSC:
Primary 52A25; Secondary 53C45

DOI:
https://doi.org/10.1090/S0002-9947-1985-0776397-6

MathSciNet review:
776397

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Abstract: For a smooth closed surface in the classical total mean curvature is defined by , where are the principal curvatures at on . If is a polyhedral surface, there is a well known discrete version given by , where represents edge length and the corresponding dihedral angle along the edge. In this article formulas involving differentials of total mean curvature (closely related to the differential formula of L. Schláfli) are applied to several questions concerning Lipschitizian mappings of polyhedral surfaces.

For example, the simplest formula may be used to show that the remarkable flexible polyhedral spheres of R. Connelly must flex with constant total mean curvature. Related differential formulas are instrumental in showing that if is a distance-increasing function and , then .

This article (part I) is mainly concerned with problems in . In the sequel (part II) related questions in and , as well as , will be considered.

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DOI:
https://doi.org/10.1090/S0002-9947-1985-0776397-6

Article copyright:
© Copyright 1985
American Mathematical Society