An index formula for hemispheres of a $C^2$-regular convex closed surface in Euclidean $3$-space
Abstract
Carath\'eodory's conjecture has long been regarded as one of the central problems in the classical theory of convex surfaces. In this paper, we establish an index formula for hemispheres of convex closed surfaces under -regularity. The proof is based on studying a vertical section of the null hypersurfaces in Lorentz--Minkowski -space associated with the originally given convex surface. As a consequence, the conjecture is affirmatively solved in the -case.
Cite
@article{arxiv.2512.23181,
title = {An index formula for hemispheres of a $C^2$-regular convex closed surface in Euclidean $3$-space},
author = {Naoya Ando and Masaaki Umehara},
journal= {arXiv preprint arXiv:2512.23181},
year = {2026}
}
Comments
We found a gap in the step where we claimed that, for $k$ sufficiently close to $1$, all $k$-umbilics are contained in $\bigcup_{i=1}^\infty \Omega_i$. So this manuscript should not be cited. In fact, Example 4 of arXiv:0808.0851 provides a configuration that is not excluded by our argument, thereby revealing the gap