2003/02/18 by Ronaldo Garcia, Garcia, Ronaldo, Jorge Sotomayor +1
Mathematics · #34D30 #37C75 #53A05 #53C12 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #math.DG #msc:34D30 #msc:37C75 #msc:53A05 #msc:53C12
paper · pdf · doi:10.48550/arxiv.math/0302196
23 pages, 4 figures, Prepublication Laboratoire de Topologie, n. 294 (2002) Universite de Bourgogne; to appear in Bull. Bras. Math. Soc
arxiv created 2003/02/18 · openalex publication_date 2003/02/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider oriented surfaces immersed in \mathbb R3. Associated to them, here are studied pairs of transversal foliations with singularities, defined on the Elliptic region, where the Gaussian curvature \mathcal K, given by the product of the principal curvatures k1, k2 is positive. The leaves of the foliations are the lines of harmonic mean curvature, also called characteristic or diagonal lines, along which the normal curvature of the immersion is given by \mathcal K/\mathcal H, where \mathcal H=(k1+k2)/2 is the arithmetic mean curvature. That is, \mathcal K/\mathcal H=((1/k1 + 1/k2)/2)-1 is the harmonic mean of the principal curvatures k1, k2 of the immersion. The singularities of the foliations are the umbilic points and parabolic curves, where k1 = k2 and \mathcal K = 0, respectively. Here are determined the structurally stable patterns of harmonic mean curvature lines near the umbilic points, parabolic curves and harmonic mean curvature cycles, the periodic leaves of the foliations. The genericity of these patterns is established. This provides the three essential local ingredients to establish sufficient conditions, likely to be also necessary, for Harmonic Mean Curvature Structural Stability of immersed surfaces. This study, outlined towards the end of the paper, is a natural analog and complement for that carried out previously by the authors for the Arithmetic Mean Curvature and the Asymptotic Structural Stability of immersed surfaces.