发表机构
Indian Institute of Science Education and Research Pune; Indian Statistical Institute, Delhi Centre; Tata Institute of Fundamental Research（印度科学教育与研究学院浦那分校; 印度统计学院德里中心; 塔塔基础研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用海森堡-泡利-魏尔不确定性原理，研究黎曼流形的格罗莫夫-豪斯多夫稳定性与刚性，确立了不同曲率流形的收敛性，修正了现有HPW不等式并给出定量刚性估计。
AI 中文摘要
经典海森堡-泡利-魏尔（HPW）不等式在黎曼流形上呈现出强刚性现象：即在卡尔丹-阿达马流形以及具有非负里奇曲率的流形上，欧氏HPW不等式的有效性或极值函数的存在性，会严格迫使该流形等距于欧氏空间$\boldsymbol{\text{R}}^n$。这种几何差异推动了对曲率依赖修正项及其相关稳定性性质的研究。本文中，我们研究HPW不等式的几何稳定性。具体而言，给定一列带基点的黎曼流形以及适当归一化且HPW亏缺消失的函数，我们探讨该序列是否在带基点的格罗莫夫-豪斯多夫拓扑下收敛到对应的模型空间。我们证明，对于截面曲率上界为$c<0$的捏合卡尔丹-阿达马流形，这些流形收敛到模型双曲空间$\boldsymbol{\text{H}}^n_c$；在非负里奇曲率情形下，我们确立了在由序列矩项决定的度量重标因子下收敛到$\boldsymbol{\text{R}}^n$。由此我们推知，通过渐近体积比（AVR）构建的已知HPW不等式对于非等距于欧氏空间的非负里奇曲率流形是次优的。为解决这一问题，我们针对该情形引入了类似卡尔丹-阿达马情形的曲率修正HPW不等式。最后，我们在两种曲率情形下确立了定量刚性估计，证明在高斯轮廓处计算的HPW亏缺可明确控制与对应模型空间的适当定义距离。
英文摘要
The classical Heisenberg Pauli Weyl (HPW) inequality exhibits a strong rigidity phenomenon on Riemannian manifolds i.e. on Cartan Hadamard manifolds and those with non-negative Ricci curvature, the validity of the Euclidean HPW inequality or the existence of extremizers strictly forces the manifold to be isometric to Euclidean space, $\mathbb{R}^n$. This geometric discrepancy motivates the study of curvature dependent corrections and their associated stability properties. In this article, we investigate the geometric stability of the HPW inequality. Specifically, given a sequence of pointed Riemannian manifolds and appropriately normalized functions with a vanishing HPW deficit, we address whether the sequence converges to the corresponding model space in the pointed Gromov Hausdorff topology. We prove that for pinched Cartan Hadamard manifolds with sectional curvature bounded above by $c < 0$, the manifolds converge to the model hyperbolic space $\mathbb{H}^n_c$. In the non-negative Ricci curvature setting, we establish convergence to $\mathbb{R}^n$, up to a metric rescaling factor governed by the sequence's moment term. Consequently, we deduce that the known HPW inequality formulated via the asymptotic volume ratio (AVR) is suboptimal for non negatively Ricci curved manifolds not isometric to Euclidean space. To resolve this, we introduce a curvature corrected HPW inequality for this setting, analogous to the Cartan Hadamard case. Finally, we establish quantitative rigidity estimates in both curvature regimes, demonstrating that the HPW deficit when evaluated at Gaussian profiles which explicitly controls an appropriately defined distance to the respective model space.
Comments37 pages
