2025/10/09 by Romain Speciel, Speciel, Romain
Mathematics · #Spectral Theory in Mathematical Physics #advanced mathematical theories #Differential Equations and Boundary Problems
paper · pdf · doi:10.48550/arxiv.2510.08822
The Laplacian Δ_\mathbbSn-1 on the unit sphere \mathbbSn-1⊂ ℝn has the property that it can explicitly be expressed in terms of Λ, the Dirichlet-to-Neumann map of the unit ball, as Δ_\mathbbSn-1=Λ2+(n-2)Λ. In this paper, we seek to characterize those manifolds for which such an exact relationship holds, and more generally measure the discrepancy of such a relationship holding in terms of geometric data. To this end, we obtain a stability estimate which shows that, for a smoothly bounded domain in ℝ3, if the commutator [Λ,Δ_\mathbbSn-1] is small then that domain is itself close to a ball. We then study the case of manifolds conformal to the ball, show that a relationship as above implies a radial metric structure, and discuss stability in this setting. Finally, we provide a modern exposition of Gohberg's lemma, a foundational result in microlocal analysis which we employ as a starting step for our reasoning.