Geometric rigidity from the Dirichlet-to-Neumann operator
Abstract
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.
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BibTeX
@misc{speciel2026geometric,
title = {{Geometric rigidity from the Dirichlet-to-Neumann operator}},
author = {Romain Speciel},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.05577},
eprint = {2608.05577},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.05577},
}
RIS
TY - UNPB
TI - Geometric rigidity from the Dirichlet-to-Neumann operator
AU - Romain Speciel
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2608.05577
UR - https://arxiv.org/abs/2608.05577
AB - 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.
ER -