Reflection groups and the eigenvalue problems of vibrating membranes with mixed boundary conditions (Q801469)

Let (M,g) be an n-dimensional space form of constant curvature (the Euclidean space \(R^ n\), the standard sphere \(S^ n\) or the hyperbolic space \(H^ n)\). Let \(\Omega\) be a bounded domain in M with an appropriately regular boundary \(\partial \Omega\). This paper refers to three different boundary value eigenvalue problems for the non-negative Laplacian of (M,g). \(S_ 1(\Omega)\), \(S_ 2(\Omega)\), \(S_ 3(\Omega)\) denote the spectra of these problems respectively. Two domains \(\Omega\), \({\tilde \Omega}\) are called congruent in the space form (M,g) if there exists an isometry \(\phi\) of (M,g) such that \(\phi (\Omega)={\tilde \Omega}.\) In an earlier paper [Ann. Sci. Éc. Norm. Supér., IV. Sér. 15, 441- 456 (1982; Zbl 0505.58036)] the author proved that there exist two domains \(\Omega\), \({\tilde \Omega}\) in x such that \(S_ 1(\Omega)=S_ 1({\tilde \Omega})\) and \(S_ 2(\Omega)=S_ 2({\tilde \Omega})\), but \(\Omega\) and \({\tilde \Omega}\) are not congruent in X, where \(X\equiv R^ n\) or \(X\equiv S^{n-1}\), \(n\geq 4.\) In this paper the author gives the proof of the following: Let (M,g) be simply connected with \(n\geq 4\). Then there exist two domains \(\Omega\), \({\tilde \Omega}\) in (M,g) such that \(S_ 1(\Omega)=S_ 1({\tilde \Omega})\), \(S_ 2(\Omega)=S_ 2({\tilde \Omega})\) and \(S_ 3(\Omega)=S_ 3({\tilde \Omega})\) for each real number \(\rho\), but \(\Omega\) and \({\tilde \Omega}\) are not congruent in (M,g).