1985/04/01 by Ralph A. Alexander · 13 citations
Mathematics · Engineering · Computer Science · #Geometric Analysis and Curvature Flows #Advanced Numerical Analysis Techniques #Contact Mechanics and Variational Inequalities #Mathematics #Mean curvature #Combinatorics #Curvature #Surface (topology) #Geometry #Principal curvature #Dihedral angle #Mathematical analysis #Sigma #Mean curvature flow #Differential geometry #Physics
paper · doi:10.2307/1999957
published in Transactions of the American Mathematical Society 288(2), 661 (American Mathematical Society)
openalex publication_date 1985/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
For a smooth closed surface C in E3 the classical total mean curvature is defined by M(C) = \frac 1 2∫ (κ 1 + κ 2) dσ (p), where κ 1,κ 2 are the principal curvatures at p on C. If C is a polyhedral surface, there is a well known discrete version given by M(C) = \frac 1 2Σ li(π - α i), where li represents edge length and α i 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 Σ li dα i = 0 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 f: E2 → E2 is a distance-increasing function and K ⊂ E2, then \operatorname Per(\operatorname conv K) \leqslant \operatorname Per(\operatorname conv f[K]). This article (part I) is mainly concerned with problems in En. In the sequel (part II) related questions in Sn and Hn, as well as En, will be considered.