An indefinite Coulomb interaction from the Steklov spectrum of perforated manifolds

Zhonggan Huang; Raghavendra Venkatraman

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arXiv · 2607.25211

An indefinite Coulomb interaction from the Steklov spectrum of perforated manifolds

Zhonggan Huang·Raghavendra Venkatraman

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Abstract

We establish optimal convergence rates for Steklov eigenvalues and harmonically extended eigenfunctions toward their weighted Laplace--Beltrami counterparts on a closed manifold perforated by many small geodesic balls. The holes have radii that scale critically with respect to their spacing in the sense that the boundary area of each hole balances with the weighted volume of its Voronoi cell. We then derive a higher-order expansion of the Steklov eigenvalues; in dimensions two and three, we identify two correction scales. The expansion is governed by an indefinite Coulomb-type energy of the discrepancy between the boundary and bulk measures, mediated by the reduced Green function of the limiting operator. The proof relies on sharp estimates for certain auxiliary functions that we introduce in order to quantify the discrepancy measure between the surface measure on the holes and their background density. Our paper serves to bridge an emerging literature in spectral geometry with one on systems of points interacting via Coulomb-type energies.

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Zhonggan Huang, Raghavendra Venkatraman. 2026-07-28. An indefinite Coulomb interaction from the Steklov spectrum of perforated manifolds. https://arxiv.org/abs/2607.25211

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