Spectral invariants for polygons and orbisurfaces

TL;DR

This thesis develops explicit formulas for heat invariants of polygons and orbisurfaces with constant curvature, revealing how spectral data encodes geometric and topological features.

Contribution

It unifies heat invariant formulas for polygons in various constant curvature spaces and extends results to orbisurfaces, linking spectral invariants to geometric and topological properties.

Findings

Heat invariants determine polygon angles and Euler characteristic.

Explicit heat invariant formulas for hyperbolic, Euclidean, and spherical polygons.

Spectral data reveals topology and singularities of orbisurfaces.

Abstract

In this thesis we deal with spectral invariants for polygons and closed orbisurfaces of constant Gaussian curvature. In each case our method is to study the heat kernel and the asymptotic expansion of the heat trace. First, we investigate hyperbolic polygons, i.e. relatively compact domains in the hyperbolic plane with piecewise geodesic boundary. We compute the asymptotic expansion of the heat trace associated to the Dirichlet Laplacian of any hyperbolic polygon, and we obtain explicit formulas for all heat invariants. Analogous results for Euclidean and spherical polygons were known before. We unify these results and deduce the heat invariants for arbitrary polygons, i.e. for relatively compact domains with piecewise geodesic boundary contained in a complete Riemannian manifold of constant Gaussian curvature. It turns out that the heat invariants provide much information about a polygon, if the curvature does not vanish. For example, then the multiset of all real angles (i.e. those which are not equal to pi) and the Euler characteristic of a polygon are spectral invariants. Furthermore, we compute the asymptotic expansion of the heat trace for any closed Riemannian orbisurface of constant curvature, and obtain explicit formulas for all heat invariants. If the curvature does not vanish, then it is possible to detect interesting information about the topology and the singular set of an orbisurface from the heat invariants.