Lines of Mean Curvature on Surfaces Immersed in R3

Abstract

Associated to oriented surfaces immersed in R^3 here are studied pairs of transversal foliations with singularities, defined on the Elliptic region, where the Gaussian curvature K, given by the product of the principal curvatures k_1, k_2 of the immersion, is positive. The leaves of the foliations are the lines of M- mean curvature, along which the normal curvature of the immersion is given by a function M = M(k_1,k_2) in [k_1, k_2], called a M- mean curvature, whose properties extend and unify those of the arithmetic H=(k_1+k_2)/2, the geometric K^{1/2} and harmonic K/H=((1/{k_1} + 1/{k_2})/2)^{-1} classical mean curvatures. The singularities of the foliations are the umbilic points and parabolic curves, where k_1 = k_2 and K = 0, respectively. Here are determined the patterns of M- mean curvature lines near the umbilic points, parabolic curves and M- mean curvature cycles (the periodic leaves of the foliations), which are structurally stable under small perturbations of the immersion. The genericity of these patterns is also established. These patterns provide the three essential local ingredients to establish sufficient conditions, likely to be also necessary, for M-Mean Curvature Structural Stability of immersed surfaces. This constitutes a natural unification and complement for the results obtained previously by the authors for the Arithmetic, Asymptotic, Geometric and Harmonic, classical cases of Mean Curvature Structural Stability.