具有有界曲率但无有界平均曲率欧几里得等距浸入的流形
Bounded curvature manifolds without Euclidean isometric immersions of bounded mean curvature
- School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文构造了任意维数下具有有界曲率和单射半径的完备黎曼度量，证明其不存在有界平均曲率的欧几里得等距浸入，从而否定了Yau问题52。
AI中文摘要：
对于每个整数$n\ge2$，我们在$\mathbb{R}^n$上构造一个光滑完备黎曼度量$G_n$，其全曲率范数至多为1且单射半径至少为1，使得不存在到有限维欧几里得空间的$C^2$等距浸入具有有界平均曲率。在二维情形，有界第二基本形式将给出共形因子拉普拉斯算子的均匀受控有限雅可比表示。我们构造不相交的共形块，其曲率保持有界而有限雅可比表示代价趋于无穷。取欧几里得乘积可得到所有更高维数。该结果在更强的有界全曲率假设下否定了Yau问题52。
英文摘要：
For every integer $n\ge2$, we construct a smooth complete Riemannian metric $G_n$ on $\mathbb{R}^n$ with full curvature norm at most one and injectivity radius at least one for which no $C^2$ isometric immersion into a finite-dimensional Euclidean space has bounded mean curvature. In dimension two, bounded second fundamental form would give uniformly controlled finite Jacobian representations of the Laplacian of the conformal factor. We construct disjoint conformal blocks for which the curvature remains bounded while the finite Jacobian representation cost tends to infinity. Taking Euclidean products gives all higher dimensions. The result answers Yau's Problem~52 negatively under the stronger assumption of bounded full curvature.
