叶状 Plateau 问题、曲面 Radon 变换与三维边界面积刚性
Foliated Plateau problems, surface Radon transforms, and boundary area rigidity in dimension three
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中文总结 AI 辅助
本文研究三维黎曼球上由椭圆曲率泛函定义的 $\Phi$-曲面，通过求解叶状 Plateau 问题构造单位切丛的叶状结构，分析曲面 Radon 变换的核与单射性，并应用于极小曲面叶状结构下的边界面积刚性问题的解决。
中文摘要 AI 辅助
本文研究三维黎曼球上的积分几何问题，其中积分在极小曲面或更一般地由椭圆曲率泛函 $\Phi$ 定义的 $\Phi$-曲面上进行。由边界上的圆所张成的所有 $\Phi$-曲面的空间是一个三维流形，我们称之为圆空间。我们证明，当度量是 $\Phi$-简单的（这一概念将测地线情形中的简单度量概念推广到该情形）时，$\Phi$-曲面的 Gauss 提升定义了单位切丛的一个叶状结构，该叶状结构应被视为标准测地线叶状结构的二维类比。这是通过求解球上的叶状 Plateau 问题实现的。然后我们分析沿曲面积分所对应的曲面 Radon 变换，并证明它具有有限维核；我们还证明它对于度量的一个开稠密集是单射的。在极小曲面叶状结构的特殊情形下，我们应用这些结果来解决以下边界面积刚性问题：极小曲面的面积集合是否在等距意义下唯一确定度量？
英文摘要
The present paper studies integral geometry problems on three-dimensional Riemannian balls, where integration is performed over minimal surfaces or, more generally, $Φ$-surfaces defined by an elliptic curvature functional $Φ$. The space of all $Φ$-surfaces spanned by round circles on the boundary is a three-dimensional manifold, which we call the space of circles. We show that, when the metric is $Φ$-simple - a notion which extends to this setting the notion of simple metrics in the geodesic case -, the Gauss lifts of the $Φ$-surfaces define a foliation of the unit tangent bundle that should be viewed as a two-dimensional analogue of the standard geodesic foliation. This is achieved by solving a foliated Plateau problem on the ball. We then analyze the associated surface Radon transform corresponding to integration along the surfaces and show that it has a finite-dimensional kernel; we also prove that it is injective for an open and dense set of metrics. In the special case of a foliation by minimal surfaces, we apply these results to solve the following boundary area rigidity problem: does the collection of areas of the minimal surfaces determine the metric up to isometry?
发表机构
- Universidad de la República（乌拉圭共和国大学）
- Université Paris-Saclay（巴黎萨克雷大学）
- University of Chicago（芝加哥大学）
- PUC-Rio（天主教里约热内卢大学）
机构由 AI 辅助整理，请以论文原文为准。
