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A Meyer–Vietoris Formula for the Determinant of the Dirichlet-to-Neumann Operator on Riemann Surfaces

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Abstract

This paper presents a Meyer–Vietoris type gluing formula for a conformal invariant of a Riemannian surface with boundary that is defined by the determinant of the Dirichlet-to-Neumann operator. The formula is used to bound the asymptotics of the invariant under degeneration. It is shown that the associated height function on the moduli space of hyperbolic surfaces with geodesic boundary is proper only in genus zero. Properness implies a compactness theorem for Steklov isospectral metrics in the case of genus zero. The formula also provides asymptotics for the determinant of the Laplacian with Dirichlet or Neumann boundary conditions. For the proof, we derive an extension of Kirchhoff’s weighted matrix tree theorem for graph Laplacians with an external potential.

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Notes

  1. Graph Laplacians will appear throughout the paper. Notation and the relevant results can be found in Sect. 6, which is independent of the other sections of this paper.

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Acknowledgements

The author is grateful to the editors for putting together this special volume. He also thanks Jean Lagacé and David Sher for their comments.

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  1. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Richard A. Wentworth

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  1. Richard A. Wentworth
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Correspondence to Richard A. Wentworth.

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Dedicated to Peter Li on the occasion of his 70th birthday.

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The author gratefully acknowledges support from NSF Grants DMS-1906403 and DMS-2204346.

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Wentworth, R.A. A Meyer–Vietoris Formula for the Determinant of the Dirichlet-to-Neumann Operator on Riemann Surfaces. J Geom Anal 33, 51 (2023). https://doi.org/10.1007/s12220-022-01097-6

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