法曲率与射影 systole
Small Normal Curvature and Three-Manifold Topology
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中文总结 AI 辅助
该研究针对实射影空间到单位球的光滑浸入，利用射影 systolic 不等式得到其最大法曲率的严格下界，重现二维情形的 Petrunin 定理并证实三维情形的首个开放问题。
中文摘要 AI 辅助
对于光滑浸入 F:ℝℙᵐ↪overline{ℬ}^N(1)，我们发现严格的 systolic 不等式对其最大法曲率 κ(F) 施加了严格下界。在维数 m=2、3 时，Pu 和 Bray–Brendle–Eichmair–Neves 的严格不等式给出 κ(F)²≥2m/(m+1)。等号仅对 Veronese 嵌入成立。这重现了ℝℙ²的 Petrunin 定理，且对ℝℙ3，证实了他关于实射影空间问题的首个开放情形。
英文摘要
For $m=2,3$, we prove that every smooth immersion $F:\mathbb{R}\mathbb{P}^m\looparrowright\overline{\mathbb B}^{,N}(1)$ satisfies $κ(F)^2\ge 2m/(m+1)$, with equality only for the Veronese embedding, up to congruence. We also prove that a closed, connected, orientable three-manifold admitting an immersion into a Euclidean unit ball with $κ(F)\le\sqrt{3/2}$ is diffeomorphic to $S^3$, $\mathbb{R}\mathbb{P}^3$, or $S^2\times S^1$. All three possibilities occur, while $κ(F)<\sqrt{3/2}$ forces $X\cong S^3$. These results answer a question of Petrunin and prove a conjecture of Chodosh--Li concerning the normal curvature of three-manifolds. The key intrinsic input is the strict scalar--systolic inequality \[ (\min_Y R_g)\text{sys}(g)^2<6π^2 \] for every spherical three-space form $Y$ with $|π_1(Y)|>2$. Its proof uses systolic monotonicity along Ricci flow with surgery. This strict inequality complements the sharp scalar--systolic inequality for $\mathbb{R}\mathbb{P}^3$ of Bray--Brendle--Eichmair--Neves.
发表机构
- Hong Kong University of Science and Technology（香港科技大学）
- Sorbonne Université（索邦大学）
机构由 AI 辅助整理，请以论文原文为准。
