欧氏球的Steklov刚性
Geometric rigidity from the Dirichlet-to-Neumann operator
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中文总结 AI 辅助
该研究针对Steklov谱的刚性问题，证明了n≥3维空间中，若有界光滑区域的Steklov谱以足够快速率趋近于球的Steklov谱则该区域必为球，且欧氏球由其Steklov谱唯一确定。
中文摘要 AI 辅助
设$\Omega\subset \mathbb{R}^n$（$n\geq 3$）是具有光滑边界的有界区域。我们证明，若$\Omega$的Steklov谱以足够快的速率趋近于某个球的Steklov谱，则$\Omega$本身必为一个球。特别地，在任意维度下，在所有具有光滑（可能不连通）边界的有界区域中，欧氏球由其Steklov谱唯一确定。
英文摘要
The analytic properties of the Dirichlet-to-Neumann operator reflect the geometry of the underlying manifold. As a guiding example of this relationship, consider the Euclidean unit ball. There, the Dirichlet-to-Neumann operator has an explicitly computable spectrum, consisting of the nonnegative integers with multiplicity, and commutes with the boundary Laplacian. In this paper, we study the extent to which these properties distinguish the ball among various classes of manifolds. We show that, in every dimension, Euclidean balls are uniquely determined by their Steklov spectrum among bounded Euclidean domains with smooth and possibly disconnected boundary, answering a well-known open question. We then classify all smooth compact connected oriented surfaces with nonempty boundary whose Dirichlet-to-Neumann operator commutes with the boundary Laplacian. Finally, in dimensions three and above, we prove that, for smooth metrics conformal to the Euclidean ball, the Dirichlet-to-Neumann operator commutes with the Laplacian on the round sphere if and only if the conformal factor is radial.
发表机构
- Stanford University（斯坦福大学）
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