Harmonic mean curvature lines on surfaces immersed in $\mathbb{R}^{3}$

Abstract.

Consider oriented surfaces immersed in $\mathbb{R}^{3}$. Associated to them, here are studied pairs of transversal foliations with singularities, defined on the Elliptic region, where the Gaussian curvature ${\cal K}$, given by the product of the principal curvatures k ₁, k ₂ 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 ${\cal K}/{\cal H}$, where ${\cal H} = {\left( {k_{1} + k_{2} } \right)}/2$ is the arithmetic mean curvature. That is, ${cal K}/{\cal H} = {\left( {{\left( {1/k_{1} + 1/k_{2} } \right)}/2} \right)}^{{ - 1}}$ is the harmonic mean of the principal curvatures k ₁, k ₂ of the immersion. The singularities of the foliations are the umbilic points and parabolic curves, where k ₁ = k ₂ and ${\cal K} = 0$, 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, [13, 14, 17], and also extended recently to the case of the Geometric Mean Curvature Configuration [15].