Yau有界平均曲率嵌入问题的一个反例
A counterexample to Yau's bounded-mean-curvature embedding question
- Fudan University（复旦大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对Yau关于有界Ricci曲率与正单射半径流形是否存在有界平均曲率等距嵌入的问题，构造了四维及以上维度的反例，证明此类嵌入不存在。
AI中文摘要：
Yau提出如下问题：是否每个具有有界Ricci曲率和正单射半径的完备黎曼流形都能等距嵌入到某个欧氏空间中，且具有有界平均曲率？我们构造了一个反例：$\u211d^4$上的一个光滑完备度量，其Ricci曲率有界，单射半径为正，但全曲率无界。高斯方程因此迫使该度量到有限维欧氏空间的任何$C^2$等距浸入都具有无界平均曲率。该度量由不相交的曲率峰组装而成，其Ricci收缩一致有界。短测地线曲率估计和Jacobi方程给出了均匀单射半径界。该构造在至少四维的每个维度中都成立，并且在四维中，Ricci曲率可以选择为在无穷远处趋于零。
英文摘要：
Yau asked whether every complete Riemannian manifold with bounded Ricci curvature and positive injectivity radius admits an isometric embedding into some Euclidean space with bounded mean curvature. We construct a counterexample: a smooth complete metric on $\mathbb{R}^4$ with bounded Ricci curvature, positive injectivity radius, and unbounded full curvature. The Gauss equation then forces every $C^2$ isometric immersion of this metric into a finite-dimensional Euclidean space to have unbounded mean curvature. The metric is assembled from disjoint curvature peaks whose Ricci contractions stay uniformly bounded. A short-geodesic curvature estimate and the Jacobi equation give the uniform injectivity-radius bound. The construction works in every dimension at least four, and in dimension four the Ricci curvature may be chosen to tend to zero at infinity.
