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Green Function and Self-adjoint Laplacians on Polyhedral Surfaces

Part of: Riemann surfaces Spectral theory and eigenvalue problems Deformations of analytic structures Partial differential equations on manifolds; differential operators

Published online by Cambridge University Press:  02 July 2019

Alexey Kokotov  and
Kelvin Lagota
Alexey Kokotov
Affiliation:
Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Québec, H3G 1M8 Email: alexey.kokotov@concordia.cakelvin.lagota@concordia.ca
Kelvin Lagota
Affiliation:
Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Québec, H3G 1M8 Email: alexey.kokotov@concordia.cakelvin.lagota@concordia.ca
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Abstract

Using Roelcke’s formula for the Green function, we explicitly construct a basis in the kernel of the adjoint Laplacian on a compact polyhedral surface $X$ and compute the $S$-matrix of $X$ at the zero value of the spectral parameter. We apply these results to study various self-adjoint extensions of a symmetric Laplacian on a compact polyhedral surface of genus two with a single conical point. It turns out that the behaviour of the $S$-matrix at the zero value of the spectral parameter is sensitive to the geometry of the polyhedron.

Keywords

self-adjoint extensionLaplacianpolyhedral surface

MSC classification

Primary: 30F30: Differentials on Riemann surfaces
Secondary: 30F45: Conformal metrics (hyperbolic, Poincaré, distance functions) 35P99: None of the above, but in this section 58J52: Determinants and determinant bundles, analytic torsion 30F10: Compact Riemann surfaces and uniformization 32G15: Moduli of Riemann surfaces, Teichmüller theory
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 5 , October 2020 , pp. 1324 - 1351
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Copyright
© Canadian Mathematical Society 2019

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