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Green Function and Self-adjoint Laplacians on Polyhedral Surfaces
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 72 / Issue 5 / October 2020
- Published online by Cambridge University Press:
- 02 July 2019, pp. 1324-1351
- Print publication:
- October 2020
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- Article
- Export citation
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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.
