发表机构
Hong Kong University of Science and Technology; Princeton University（香港科技大学; 普林斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了具有任意大Urysohn宽度和稳定极小圆盘半径的4-PIC度量，否证了正各向同性曲率下的宽度与圆盘半径界，并在闭偶维流形上证明了尖锐的谱估计及等号刚性。
AI 中文摘要
对于每个$n\ge4$和$L>0$，我们在$S^n$上构造一个光滑的$4$-PIC度量，其Urysohn $1$-宽度至少为$L$，并且包含一个内蕴内半径至少为$L$的嵌入稳定极小圆盘。这些例子否证了在各向同性曲率正下界条件下所提出的宽度和稳定圆盘半径界。在闭偶维流形上，我们在$\sigma$-PIC条件下证明了尖锐估计$\lambda_1^{(2)}\ge(n-1)\sigma/2$，并表明若达到该界的闭特征形式在某点秩至少为四，则等号迫使流形为圆形的。
英文摘要
For every $n\ge4$ and $L>0$, we construct a smooth $4$-PIC metric on $S^n$ with Urysohn $1$-width at least $L$ and an embedded stable minimal disk of intrinsic inradius at least $L$. These examples disprove the proposed width and stable-disk radius bounds under a positive lower bound for isotropic curvature. On closed even-dimensional manifolds, we prove the sharp estimate $λ_1^{(2)}\ge(n-1)σ/2$ under $σ$-PIC and show that equality forces roundness if a closed eigenform attaining the bound has rank at least four at some point.
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