负曲率曲面的有界面积类
The bounded area class of negatively curved surfaces
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中文总结 AI 辅助
该研究针对带挤压负曲率的定向曲面，证明其面积形式在二阶有界上同调中（除圆盘、圆柱外）非平凡，还揭示有界面积类对紧致曲面可识别常曲率度量、对无限型曲面无法区分非等距双曲结构的特性。
中文摘要 AI 辅助
设S为定向曲面，可能具有无限型，赋予其带挤压负曲率的完备黎曼度量。我们证明，除非S微分同胚于圆盘或圆柱，否则面积形式在S的二阶有界上同调群中定义了一个非平凡类。这与n维（n>2）情形形成鲜明对比，后者中体积形式在有界上同调中的非平凡性与流形的Cheeger常数相关。我们还讨论有界面积类如何依赖于度量：对于紧致曲面，它在挤压负曲率度量中识别常曲率度量；而即使在无限型曲面情形，它也无法区分非等距的双曲结构。更确切地说，当S紧致时，我们证明在固定面积的负曲率结构中，常曲率结构是有界面积类范数的唯一极小值点。
英文摘要
Let $S$ be an oriented surface, possibly of infinite type, endowed with a complete Riemannian metric with pinched negative curvature. We prove that the area form defines a non-trivial class in the second bounded cohomology group of $S$, unless $S$ is diffeomorphic to the disc or the cylinder. This is in sharp contrast with the $n$-dimensional case, $n>2$, where the non-triviality of the volume form in bounded cohomology is related to the Cheeger constant of the manifold. We also discuss how the bounded area class depends on the metric: we prove that, for compact surfaces, it recognizes constant curvature metrics among pinched negatively curved ones, while, even in the case of surfaces of infinite type, it does not distinguish non-isometric hyperbolic structures. More precisely, when $S$ is compact we show that, among the negatively curved structures of fixed area, the ones with constant curvature provide the unique minimizers for the norm of the bounded area class.
发表机构
- Dipartimento di Matematica（数学系）
- Scuola Normale Superiore（比萨高等师范学院）
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