Dimensional asymptotics of determinants on \(S^n\), and proof of Bär-Schopka's conjecture (Q957891)

For the standard metric on the \(n\)-dimensional sphere \(S^n_{\text{can}}\), the author investigates the functional determinant \(\det(D,S^n_{\text{can}})\), in particular the asymptotics for \(n \to \infty\). As a corollary of the results it is shown that \[ \lim_{n\to \infty}\det (D,S^n_{\text{can}})=1 \] which was conjectured by \textit{C. Bär} and \textit{S. Schopka} [in: Hildebrandt, Stefan (ed.) et al., Geometric analysis and nonlinear partial differential equations. Berlin: Springer. 39--67 (2003; Zbl 1049.58030)]. Further results are presented for operators \(\Delta +\beta R,\) where \(\Delta\) is the Laplacian, \(\beta\) is a real constant and \(R\) is the scalar curvature.