欧几里得 3 空间中 $C^2$ 正规凸闭曲面的半球的指数公式
微分几何
2026-01-06 v3
摘要
Carathéodory 的猜想长期被视为经典凸曲面理论的中心问题之一。本文在 正规性下，建立凸闭曲面半球的指数公式。该证明基于研究与原给定凸曲面相关的洛伦兹-關可夫斯基 4 空间中零面谱的垂直截面。作为结果， 情况下的猜想得到肯定的解答。
引用
@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}
}
备注
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