发表机构
Institute of Software, Chinese Academy of Sciences; Department of Applied Mathematics, Xi’an Jiaotong-Liverpool University（中国科学院软件研究所; 西交利物浦大学应用数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究黎曼流形上截面曲率与Ricci曲率符号在逐点夹逼条件下能否共存，通过支撑刚性定理证明尖锐阈值，并给出显式构造与紧化障碍。
AI 中文摘要
我们研究连通的黎曼流形，其上每一点的截面曲率或 Ricci 张量要么严格为正、严格为负、要么为零，并询问在逐点夹逼条件下这两种符号能否共存。一个关于半正定无散度对称张量的一般支撑刚性定理是共同的分析机制。对于维数 $n\ge 3$ 的截面曲率，绝对逐点夹逼比的任何局部一致正下界都排除符号变化。当夹逼常数 $\delta_0$ 固定时，平坦集没有 $C^1$ 超曲面片；当 $\delta_0>1/2$ 时平坦集为空，当 $\delta_0=1/2$ 时平坦集是局部多孔的。这些结论在多种意义上是尖锐的：存在光滑的共形平坦局部度量，当夹逼退化时，其截面曲率在平坦超曲面两侧改变符号；并且球面上存在具有孤立平坦点的闭的恰好 $1/q$ 夹逼度量。对于 Ricci 曲率，尖锐阈值是 \\[ \delta_c=\frac1{n-1}. \\] 局部一致间隙 $\delta_{\mathrm{Ric}}>\delta_c$ 迫使全局只有一个 Ricci 符号。反之，对于每个 $0<\delta<\delta_c$，存在具有精确逐点夹逼 $\delta_{\mathrm{Ric}}\equiv\delta$ 的局部变号度量，并且在 $\delta=\delta_c$ 处也存在精确的局部例子。如果间隙在 Ricci 平坦界面处塌缩，则逐点严格性 $\delta_{\mathrm{Ric}}>\delta_c$ 是不够的。对于每个亚临界 $\delta$，我们给出 $\mathbb{S}^1\times\mathbb{S}^{n-1}$ 上的显式闭度量，以及到 $\mathbb{R}\times\mathbb{S}^{n-1}$ 的完整周期提升，其最优全局下夹逼常数恰好为 $\delta$。最后，我们证明了两个特定于 ansatz 的障碍，阻止精确局部构造的紧化。证明的主要内容由 ChatGPT 5.6 sol 生成，并由作者验证。
英文摘要
We study connected Riemannian manifolds on which either the sectional curvature or the Ricci tensor is, at each point, strictly positive, strictly negative, or zero, and ask whether the two signs can coexist under pointwise pinching. A general support-rigidity theorem for positive semidefinite divergence-free symmetric tensors is the common analytic mechanism. For sectional curvature in dimension $n\ge 3$, any locally uniform positive lower bound for the absolute pointwise pinching ratio rules out a change of sign. With a fixed pinching constant $δ_0$, the flat set has no $C^1$ hypersurface piece; it is empty when $δ_0>1/2$, and is locally porous when $δ_0=1/2$. These conclusions are sharp in several senses: there are smooth conformally flat local metrics whose sectional curvature changes sign across a flat hypersurface when the pinching degenerates, and there are closed exactly $1/q$-pinched metrics on spheres with isolated flat points. For Ricci curvature, the sharp threshold is \[ δ_c=\frac1{n-1}. \] A locally uniform gap $δ_{\mathrm{Ric}}>δ_c$ forces one Ricci sign globally. Conversely, for every $0<δ<δ_c$ there are local sign-changing metrics with exact pointwise pinching $δ_{\mathrm{Ric}}\equivδ$, and exact local examples also exist at $δ=δ_c$. Pointwise strictness $δ_{\mathrm{Ric}}>δ_c$ is insufficient if the gap collapses at a Ricci-flat interface. For every subcritical $δ$ we give explicit closed metrics on $\mathbb{S}^1\times\mathbb{S}^{n-1}$, and complete periodic lifts to $\mathbb{R}\times\mathbb{S}^{n-1}$, whose optimal global lower pinching constant is exactly $δ$. Finally, we prove two ansatz-specific obstructions to compactifying the exact local constructions. The main content of the proof is generated by ChatGPT 5.6 sol and verified by the authors.
Comments25 pages. Generated by ChatGPT 5.6 sol, and verified and revised by the authors
