2006/08/18 by Miguel Abreu, Emily Dryden, Abreu, Miguel +7
Engineering · Mathematics · #55N91 (Secondary) #58J50 (Primary) 53D20 #Differential Geometry (math.DG) #FOS: Mathematics #Finite Group Theory Research #Mathematics and Applications #Spectral Theory (math.SP) #graph theory and CDMA systems #math.DG #math.SP #msc:53D20 #msc:55N91 #msc:58J50
paper · pdf · doi:10.48550/arxiv.math/0608462
23 pages, 1 figure
arxiv created 2006/08/18 · openalex publication_date 2006/08/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Which properties of an orbifold can we ``hear,'' i.e., which topological and geometric properties of an orbifold are determined by its Laplace spectrum? We consider this question for a class of four-dimensional Kähler orbifolds: weighted projective planes M:=\C P2(N1,N2,N3) with three isolated singularities. We show that the spectra of the Laplacian acting on 0- and 1-forms on M determine the weights N1, N2, and N3. The proof involves analysis of the heat invariants using several techniques, including localization in equivariant cohomology. We show that we can replace knowledge of the spectrum on 1-forms by knowledge of the Euler characteristic and obtain the same result. Finally, after determining the values of N1, N2, and N3, we can hear whether M is endowed with an extremal Kähler metric.