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Estimates and extremals for zeta function determinants on four-manifolds. (English) Zbl 0761.58053

The authors study the properties of the natural operators \(L\) (the conformal Laplacian) and \(\nabla^ 2\) (the square of the Dirac operator) on Riemannian spaces of dimension four. The main results are:
Theorem 1.1. Suppose \((M,g_ 0)\) is a 4-dimensional compact locally symmetric, Einstein manifold which is neither the standard 4-sphere nor a hyperbolic space form. Consider a conformal metric \(g_ w=e^{2w}g_ 0\). There exist constants \(C=C_ L(\text{vol}(g_ w),\zeta_ L'(0))\) and \(C=C_{V^ 2}(\text{vol}(g_ w),\zeta_{V^ 2}'(0))\) such that \(\| w\|_{2,2}\leq C\). In case \((M,g_ 0)\) is the standard 4- sphere, there is a suitable conformal transformation \(\varphi\) in the conformal diffeomorphism group \({\mathcal C}(S^ 4,g_ 0)\) so that \(\| w\|_{2,2}\leq C\) holds for the transformed conformal factor \(w_ \varphi\) given by \(e^{2w_ \varphi} g_ 0=\varphi^*(e^{2w} g_ 0)\).
Theorem 1.2. On the standard 4-sphere \((S^ 4,g_ 0)\), the standard metric \(g_ 0\) minimizes \(\log\text{det}(L)\) as well as \(- \log\text{det}(\nabla^ 2)\) among all conformal metrics of fixed volume.
Reviewer: V.Oproiu (Iaşi)

MSC:

58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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