Magnabosco Mattia, Rossi Tommaso
Institut für Angewandte Mathematik, Universität Bonn, Bonn, Germany.
Calc Var Partial Differ Equ. 2023;62(4):123. doi: 10.1007/s00526-023-02466-x. Epub 2023 Mar 20.
The Lott-Sturm-Villani curvature-dimension condition provides a synthetic notion for a metric measure space to have curvature bounded from below by and dimension bounded from above by . It was proved by Juillet (Rev Mat Iberoam 37(1), 177-188, 2021) that a large class of sub-Riemannian manifolds do not satisfy the condition, for any and . However, his result does not cover the case of almost-Riemannian manifolds. In this paper, we address the problem of disproving the condition in this setting, providing a new strategy which allows us to contradict the one-dimensional version of the condition. In particular, we prove that 2-dimensional almost-Riemannian manifolds and strongly regular almost-Riemannian manifolds do not satisfy the condition for any and .
洛特 - 施图姆 - 维拉尼曲率 - 维数条件为度量测度空间提供了一种综合概念，使其曲率下界有界于 且维数上界有界于 。朱利耶（《伊比利亚美洲数学评论》37(1)，177 - 188，2021）证明了对于任意的 和 ，一大类次黎曼流形不满足 条件。然而，他的结果并未涵盖几乎黎曼流形的情况。在本文中，我们解决了在这种情况下否定 条件的问题，提供了一种新策略，使我们能够反驳 条件的一维版本。特别地，我们证明了二维几乎黎曼流形和强正则几乎黎曼流形对于任意的 和 都不满足 条件。
