A simple construction of smooth counterexamples to the Loewner and Carathéodory conjectures

Alpöge recently announced smooth counterexamples to the Loewner and Carathéodory conjectures. We present a simpler construction based on a trigonometric function as a seed. Using this, for every positive integer $k$, we construct a smooth real-valued function on the plane, flat at the origin, whose Hessian eigenline fields have an isolated singularity of index $1+k/2$. A support-function parametrization realizes this index as an umbilic index of a non-real analytic $C^\infty$-regular surface germ in Euclidean $3$-space. The same framework, applied to a rapidly decreasing function on the whole plane, gives a strictly convex closed smooth surface with exactly one umbilic, of index $2$. This paper is self-contained with regard to the existence of an umbilic of index $2$ and the construction of a counterexample to the Carathéodory conjecture.