Degenerate Umbilic Points of Analytic Surfaces

Abstract

Umbilics are points of a surface embedded in three space where normal curvatures are independent of direction. The (in)famous Carathéodory Conjecture states that a compact simply connected embedded surface has at least two umbilic points. A counterexample to this conjecture would be a surface whose principal foliation has index two at a single umbilic. All (purported) proofs of the Carathéodory Conjecture are based on analyses of the index of an umbilic, concluding that it is at most one. This investigation gives a much simpler geometric argument that the index of an umbilic on an analytic surface cannot be an integer larger than one, providing new insight into the Carathéodory Conjecture. The results also establish lower bounds for the index of an umbilic based on its degeneracy.

Figures (4)

• Figure 1: The index $I_0(a)$ for parameter $a = (1,1,1,0,0,0,1,0,1)$. The left panel shows 1000 solutions of $R(0.1 \cos(2\pi\theta),0.1 \sin(2\pi\theta),\cos(2\pi\psi),\sin(2\pi\psi)) = 0$. The full set of solutions is a curve that winds once around both axes of the torus $T^2$ with positive orientation, so the index is 1. (Note that the angular coordinate of the principal directions is chosen in the interval $[-1/4,1/4]$, corresponding to principal vectors $(\xi, \eta)$ with $\xi \ge 0$. The jumps in value of $\psi$ from $1/4$ to $-1/4$ are not discontinuities in the line field.) The right panel plots 40 lines of curvature that intersect the blue circle in equally spaced points. This also illustrates that the index is 1 because all of the lines of curvature are transverse to the circle.
• Figure 2: The index of the umbilic point at the origin jumps as umbilic points merge with the origin. The upper plots have $a_0 = 1/28, \, a_2 = 1$ where a merger occurs. When $a_2 < 1$ there two star umbilics on the $x$ axis. These are located at $x = \pm0.1$ and marked by large black dots in the lower right figure. The left plots are graphs of angular coordinates of principal vectors along the circles displayed in the right plots. The winding number along the blue curves is $0$ in both upper and lower plots, as evident that the graphs are not onto. The winding number of the green curve is $1$, and that is the index of the origin in the lower right plot. The winding number of the magenta curve is $-1/2$, the index of the star umbilic at $(x,y) = (0.1,0)$
• Figure 3: Symmetric bifurcation producing jump in index from -3 to +1 at the origin. The angular coordinates in the left hand plots are the same as those in Figure \ref{['umbi_fig_2']}.
• Figure 4: Bifurcations of a principal foliation at a degenerate umbilic of index $1$ that splits into two lemon umbilics of index $\frac{1}{2}$.