On the spectral rigidity of Hopf manifolds
If \(M\) is a compact Riemannian manifold, let \(\text{spec} (M) := \{0 = \lambda_0 < \lambda_1 \leq \lambda_2 \dots\}\) be the spectrum of the scalar Laplacian where each eigenvalue is repeated according to its multiplicity. Let \(\alpha = e^{-2\pi (b - \sqrt{-1} a)}\) for \(b > 0\). Multiplication by \(\alpha\) defines a fixed point free holomorphic action of the cyclic group on \(C^m - \{0\}\). The metric \(dz_i \circ d\overline{z}_i/|z|^2\) is invariant under this action and descends to a metric on the quotient \(M^m_\alpha\); this is the Hopf manifold. The author characterizes \(M_\alpha^m\) by its spectrum in certain cases if \(m = 2\) or if \(m =3\): Theorem. (1) Let \(m = 2\) and let \(2b\sqrt {5} > 1\). If \((M,J, g)\) is a compact Hermitian manifold with non-positive Euler-Poincaré characteristic and if \(\text{spec}(M,g) = \text{spec}(M^2_\alpha)\), then \((M,g)\) is isometric to \(M^2_\alpha\). (2) Let \(m = 3\) and let \(2b\sqrt{7} > 1\). If \((M, J, g)\) is a compact locally conformal Kähler manifold and if \(\text{spec} (M,g) = \text{spec} (M^3_\alpha)\), then \((M,g)\) is isometric to \(M^3_\alpha\).
scientific article; zbMATH DE number 920814
scientific article