A simple construction of smooth counterexamples to the Loewner and Carathéodory conjectures
Abstract
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∞-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.
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BibTeX
@misc{ando2026simple,
title = {{A simple construction of smooth counterexamples to the Loewner and Carathéodory conjectures}},
author = {Naoya Ando and Masaaki Umehara},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34036},
eprint = {2610.34036},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34036},
}
RIS
TY - UNPB
TI - A simple construction of smooth counterexamples to the Loewner and Carathéodory conjectures
AU - Naoya Ando
AU - Masaaki Umehara
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34036
UR - https://arxiv.org/abs/2610.34036
AB - 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^∞-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.
ER -