Title: Spectral Determinant on Euclidean Isosceles Triangle Envelopes
Abstract: We study extremal properties of the determinant of Friedrichs selfadjoint Laplacian on the Euclidean isosceles triangle envelopes of fixed area as a function of angles. We deduce an explicit closed formula for the determinant. Small-angle asymptotics show that the determinant grows without any bound as an angle of a triangle envelope goes to zero. We prove that the equilateral triangle envelope (the most symmetrical geometry) always gives rise to a critical point of the determinant and finds the critical value. When the area is below a certain value (approximately 1.92), the equilateral envelope minimizes the determinant. When the area exceeds this value, the equilateral envelope is a local maximum of the determinant.