Laplace eigenvalues on regular polygons: A series in 1/N

For regular polygons PN inscribed in a circle, the eigenvalues of the Laplacian converge as N→∞ to the known eigenvalues on a circle. We compute the leading terms of λN/λ in a series in powers of 1/N, by applying the calculus of moving surfaces to a piecewise smooth evolution from the circle to the polygon. The O(1/N2) term comes from Hadamardʼs formula, and reflects the change in area. This term disappears if we “transcribe” the polygon, scaling it to have the same area as the circle.