Green Function and Self-adjoint Laplacians on Polyhedral Surfaces

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.

Three-Dimensional Interpretation

This chapter explains the 3-dimensional interpretation of the plaid model. Section 5.2 stacks the blocks on top of each other in such a way that remotely adjacent blocks appear actually adjacent to each other in the stack. Section 5.3 shows how to modify the spacetime diagrams constructed in Section 4.3 and 4.4 so that they are unions of embedded loops, much like the plaid polygons. This modification is called pixilation. Section 5.4 shows that the plaid model construction and the pixilation processes are compatible with each other. Section 5.5 uses the compatibility of all the constructions to create polyhedral surfaces which simultaneously interpolate between the plaid polygons and the pixelated spacetime diagrams. These surfaces are viewed as spacetime diagrams for the plaid polygons; they are called spacetime plaid surfaces. Finally, Section 5.6 indulges in some discussion and speculation.