2019/10/31 by Victor Kalvin · 1 citation
Mathematics · Physics and Astronomy · #Analytic and geometric function theory #Boundary (topology) #Conical surface #Curvature #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Gravitational singularity #Laplace operator #Mathematical analysis #Mathematics #Metric (unit) #Polyhedron #Pure mathematics #Riemann hypothesis #Riemann surface #Type (biology) #math-ph #math.AP #math.DG #math.FA #math.MP #math.SP #msc:14H81 #msc:30F45 #msc:47A10 #msc:58J52
paper · pdf · doi:10.1016/j.jfa.2020.108866
openalex created_date 2019/10/10 · openalex publication_date 2020/11/18 · arxiv created 2020/11/27 · arxiv updated 2020/12/01 · openalex updated_date 2026/08/05
We present and prove Polyakov-Alvarez type comparison formulas for the determinants of Friederichs extensions of Laplacians corresponding to conformally equivalent metrics on a compact Riemann surface with conical singularities. In particular, we find how the determinants depend on the orders of conical singularities. We also illustrate these general results with several examples: based on our Polyakov-Alvarez type formulas we recover known and obtain new explicit formulas for determinants of Laplacians on singular surfaces with and without boundary. In one of the examples we show that on the metrics of constant curvature on a sphere with two conical singularities and fixed area 4π the determinant of Friederichs Laplacian is unbounded from above and attains its local maximum on the metric of standard round sphere. In another example we deduce the famous Aurell-Salomonson formula for the determinant of Friederichs Laplacian on polyhedra with spherical topology, thus providing the formula with mathematically rigorous proof.