Geometry and Topology Seminar (1/2)

What geometric information can one hear from the heat trace of a planar region? The classical McKean–Singer expansion recovers the area, boundary length, and Euler characteristic of a smoothly bounded domain. For domains with corners, each vertex contributes a term depending on the opening angle. These contributions are well understood when corners are "straight" (locally isometric to a Euclidean sector) but when the boundary curvature is nontrivial all the way down to a corner, much less is known.

In joint work with David Sher, we compute the t^{1/2} coefficient in the short-time heat trace expansion for planar curvilinear polygons, with both Dirichlet and Neumann boundary conditions. This coefficient splits into a boundary integral of the squared curvature of the boundary and a sum of local corner contributions, each depending only on the interior angle and the limiting one-sided curvatures of the adjacent arcs. Using a conformal model and a parametrix construction on the sector heat space, we characterize this contribution and compute it explicitly for right angles. As an application, we show that any admissible curvilinear polygon Dirichlet isospectral to a polygon must itself be a polygon, extending a result of Enciso and Gómez-Serrano beyond the straight-corner case.