On the multiplicity of umbilic points (Q2171875)

Umbilic points on a surface \(M\) in the Euclidean space \(\mathbb R^3\) or in the Minkowski space \(\mathbb R_1^3\) are the singular points of the lines of principal curvature. The Carathéodory conjecture states that any sufficiently smooth, convex and closed surface in \(\mathbb R^3\) has at least two umbilic points. This conjecture is still open, while the corresponding conjecture in \(\mathbb R_1^3\) is true, as the second author proved in [J. Math. Soc. Japan 65, No. 3, 723--731 (2013; Zbl 1278.53017)]. In the present paper the authors introduce an invariant of a surface \(M\) in \(\mathbb R^3\) or \(\mathbb R_1^3\) at an umbilic point and call it the \textit{multiplicity} of the umbilic point. The multiplicity counts the maximum number of stable umbilic points that can appear under small local deformations of the surface.