维数不超过18的浸入环面的法曲率
Normal curvature of immersed tori of dimension at most $18$
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中文总结 AI 辅助
该研究证明维数不超过18的浸入n维环面在单位球中存在点满足法曲率平方的球面积分下界，回答了Petrunin的问题，采用标量曲率障碍与共形拉普拉斯论证完成证明。
中文摘要 AI 辅助
我们证明：若n维环面Tⁿ光滑浸入于单位球Bᵠ⊂ℝᵠ且n≤18，则存在一点使得|II(u,u)|²的球面积分至少为3n/(n+2)。这回答了Petrunin在该维度下的问题，证明结合了环面的标量曲率障碍与共形拉普拉斯论证。
英文摘要
For $n\leq 18$, we prove that any smooth immersion of the $n$-torus into the closed unit ball in $\mathbb R^q$ has a point at which the spherical average of $\lvert II(v,v)\rvert^2$ is at least $3n/(n+2)$. This answers a question of Petrunin in these dimensions. The proof combines the scalar curvature obstruction for the torus with a conformal Laplacian argument, reducing the problem to a one-dimensional differential inequality. We also show that this reduction cannot yield the result in dimensions $n\geq19$.
发表机构
- Georgia Institute of Technology（佐治亚理工学院）
机构由 AI 辅助整理，请以论文原文为准。
