正负曲率条件下的度量延拓问题
The metric extension problem under positive and negative curvature conditions
浏览论文内容
中文总结 AI 辅助
本文研究正负曲率条件下的度量延拓问题，证明了紧流形上边界度量的延拓定理，并应用于球面的严格凸等距嵌入。
中文摘要 AI 辅助
我们首先证明，紧流形上的任意光滑边界度量都可以延拓为具有任意指定的 Ricci 曲率正下界的度量，而在更强的正第 k 阶中间 Ricci 曲率条件下，延拓可能因局部障碍而失败。对于负截面曲率，我们识别出一个延拓的全局障碍。接着我们考虑一类紧流形，其定义是存在一个具有负截面曲率且边界指标至多为 1 的度量。在此类中的每个流形上，我们证明任意光滑边界度量都可以延拓为具有负截面曲率且边界为严格凸脐状边界的度量。我们进一步证明，每个这样的初始度量都允许一个完备的渐近双曲等距延拓，满足相同的曲率条件并具有任意指定的共形无穷远。这类流形在边界连通和下是封闭的。作为这些延拓结果的几何应用，\\(\mathbb S^n\\) 上的任意光滑度量都允许一个严格凸的等距嵌入到配备完备负截面曲率度量的 \\(\mathbb R^{n+1}\\) 中。证明结合了颈部构造与用于上曲率界的角平滑技术。
英文摘要
We first prove that every smooth boundary metric on a compact manifold extends to a metric with any prescribed positive lower bound for Ricci curvature, whereas extensions under stronger positive \(k^{\mathrm{th}}\)-intermediate Ricci curvature conditions may fail due to local obstructions. For negative sectional curvature, we identify a global obstruction to extension. We then consider the class of compact manifolds defined by the existence of a metric with negative sectional curvature and boundary index at most one. On every manifold in this class, we prove that any smooth boundary metric extends to a metric with negative sectional curvature and strictly convex umbilical boundary. We further show that every such initial metric admits a complete asymptotically hyperbolic isometric extension with the same curvature condition and any prescribed conformal infinity. This class of manifolds is closed under boundary connected sums. As a geometric application of these extension results, every smooth metric on \(\mathbb S^n\) admits a strictly convex isometric embedding into \(\mathbb R^{n+1}\) equipped with a complete metric of negative sectional curvature. The proofs combine neck constructions with corner smoothing for upper curvature bounds.
发表机构
- Nankai University（南开大学）
- Westlake University（西湖大学）
机构由 AI 辅助整理，请以论文原文为准。
