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

Abstract

We study extremal properties of the determinant of Friederichs selfadjoint Laplacian on the Euclidean isosceles triangle envelopes of fixed area as a function of angles. Small-angle asymptotics show that the determinant grows without any bound as an angle of 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 find the critical value. Moreover, if the area of envelopes is not too large, then the determinant achieves its absolute minimum only on the equilateral triangle envelope and there are no other critical points, whereas for sufficiently large area the equilateral triangle envelope corresponds to a local maximum of the determinant.

arXiv comment27 pages, 6 figures

References in corpus (10)

• Effective action in spherical domains
• On Functional Determinants of Laplacians in Polygons and Simplices
• Further results on Functional Determinants of Laplacians in Simplicial Complexes
• Lowest Landau level on a cone and zeta determinants
• Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities
• On Determinants of Laplacians on Compact Riemann Surfaces Equipped with Pullbacks of Conical Metrics by Meromorphic Functions
• Spectral Determinants on Mandelstam Diagrams
• Representations for the derivative at zero and finite parts of the Barnes zeta function
• Polyakov formulas for conical singularities in two dimensions
• Flat conical Laplacian in the square of the canonical bundle and its regularized determinants

Cited by in corpus (3)

• Determinants of Laplacians for constant curvature metrics with three conical singularities on 2-sphere
• Polyakov formulas for conical singularities in two dimensions
• The variation of Barnes and Bessel zeta functions