Correction to: A Polyakov Formula for Sectors

Abstract

Let S α denote a finite circular sector of opening angle α ∈ (0, π) and radius one, and let e −t α denote the heat operator associated to the Dirichlet extension of the Laplacian. Based on recent joint work [2] and [3], we discovered an extra contribution to the variational Polyakov formula in [1] coming from the curved boundary component of the sector. Theorems 3 and 4 of [1] should have an added term + 14π . This calculation will appear in [2]. The corrected statements of these theorems are given below.

Spectral determinant on Euclidean isosceles triangle envelopes of fixed area as a function of angles: absolute minimum and small-angle asymptotics

“…We only note that the celebrated partially heuristic Aurell-Salomonson formula for determinants of Laplacians on polyhedra with spherical topology [4, (50)] returns a result equivalent to the one in Proposition 3.1; for details we refer to [13,Section 3.2]. We also refer to [1] for the most recent progress towards a rigorous mathematical proof of a similar Aurell-Salomonson formula in [3], though the results and methods in [1] seem not to be directly related to ours (in the correction notice the authors refer to a pre-print which is not yet available at the time of writing this paper).…”

“…An explicit expression for det ∆ m in terms of the coordinates c, τ j , and β j on C n was found in [4] and rigorously proved in [13,Sec. 3.2]; see also [1] for the most recent progress towards a rigorous mathematical proof of a similar explicit formula for the determinant of Dirichlet Laplacian on polygons in [3].…”