Dimensional asymptotics of determinants on S n , and proof of Bär-Schopka's conjecture
Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
We study the dimensional asymptotics of the effective actions, or functional determinants, for the Dirac operator D and Laplacians Δ + β R on round S n . For Laplacians the behavior depends on "the coupling strength" β, and one cannot in general expect a finite limit of ζ′(0), and for the ordinary Laplacian, β = 0, we prove it to be +∞, for odd dimensions. For the Dirac operator, Bär and Schopka conjectured a limit of unity for the determinant (Bär and Schopka, Geometric Analysis and Nonlinear PDEs, pp. 39-67, 2003), i.e. lim det (D, Scan n)=1. n→∞ We prove their conjecture rigorously, giving asymptotics, as well as a pattern of inequalities satisfied by the determinants. The limiting value of unity is a virtue of having "enough scalar curvature" and no kernel. Thus, for the important (conformally covariant) Yamabe operator, β = (n-2)/(4(n-1)), the determinant tends to unity. For the ordinary Laplacian it is natural to rescale spheres to unit volume, since lim det(Δ, Srescaled 2k+1)=1/2πe. k→∞
Originalsprog | Engelsk |
---|---|
Tidsskrift | Mathematische Annalen |
Vol/bind | 343 |
Udgave nummer | 1 |
Sider (fra-til) | 35-51 |
Antal sider | 17 |
ISSN | 0025-5831 |
DOI | |
Status | Udgivet - 1 jan. 2009 |
ID: 201991421