AI 中文总结
该研究探讨具正标量曲率的局部共形平坦流形，通过克莱因群等工具得到宏观维数、同伦群等约束，证明三维相关空间可缩性，构造了不满足欧氏刚性的反例。
AI 中文摘要
我们利用发展像的共形边界，研究具有非负或正标量曲率（PSC）的局部共形平坦（LCF）黎曼流形。对于闭定向的n维LCF PSC流形，当其基本群无限且n≥5时，我们将其黎曼万有覆盖的宏观维数限定为⌊(n-1)/2⌋，建立了中间维数以上非平凡同伦群的存在性，且在温和附加假设下，将其克莱因群极限集的豪斯多夫维数限定在区间(1, (n-2)/2)内。特别地，不存在闭的非球面流形能容许具有PSC的LCF度量。若标量曲率至少为n(n-1)且流形不等距于圆球面，则当克莱因群为初等群时，所有非零次数的光滑球面映射均在某处扩张；而当基本群无限时，发展映射均在某处扩张。我们证明，在三维空间中，对于有限基本群及S²×S¹，具有PSC的LCF度量空间及其模空间是可缩的；对于n≥4维同胚于球面空间形式但不同胚于Sⁿ的光滑流形，其模空间为空或可缩。对于具有非负标量曲率的完备开单连通LCF流形，在无穷远处附加拓扑假设时（三维时仅需第二同调消失），我们得到欧氏刚性结论。我们还证明，在n≥4维时，若不满足上述两个附加拓扑假设，欧氏结论可能不成立：我们构造了完备可缩的PSC例子，其不同胚于ℝⁿ。
英文摘要
We study locally conformally flat (LCF) Riemannian manifolds with nonnegative or positive scalar curvature (PSC), using the conformal boundary of the developing image. For closed oriented LCF $n$-manifolds with PSC and infinite fundamental group, $n\ge5$, we bound the macroscopic dimension of their Riemannian universal covers by $\lfloor(n-1)/2\rfloor$, establish the existence of a nontrivial homotopy group above the middle dimension, and, under mild additional hypotheses, bound the Hausdorff dimension of the limit sets of their Kleinian groups in the interval $(1,(n-2)/2)$. In particular, no closed aspherical manifold admits an LCF metric with PSC. If the scalar curvature is at least $n(n-1)$ and the manifold is not isometric to the round sphere, then every smooth nonzero-degree map to the sphere expands somewhere when its Kleinian group is elementary, while the developing map expands somewhere whenever the fundamental group is infinite. We prove that the space of LCF metrics with PSC and its moduli space are contractible in dimension three for finite fundamental group and for $S^2\times S^1$, and that the moduli space is empty or contractible for smooth manifolds homeomorphic to spherical space forms but not diffeomorphic to $S^n$ in dimensions $n\ge4$. For complete open simply connected LCF manifolds of nonnegative scalar curvature, we obtain Euclidean rigidity under additional topological hypotheses at infinity (and, in dimension three, from vanishing second homology alone). We also show that, in dimensions $n\ge4$, the Euclidean conclusion can fail when neither of the two additional topological hypotheses is assumed: we construct complete contractible examples of PSC that are not homeomorphic to $\mathbb{R}^n$.
