Discontinuity of geometric expansions
The author proves the following result: Let \(0=\lambda_0< \lambda_1\leq \lambda_2\leq\dots\) be any given sequence of real numbers and \(M\) a closed manifold of dimension \(p\), \(p\geq3\). Then, for any \(V\in(0,\infty)\), \(S\in(-\infty,+\infty)\), there is a sequence of metrics \((g_n)_{n\in\mathbb{N}}\) such that for each arbitrarily but fixed \(m\) we have: (i) \(\lambda_k(M,g_m)= \lambda_k\) for each \(k\leq m\), (ii) \(\text{Vol}(M,g_m)=V\), (iii) \(\int_M\text{Scal} (g_m)dV_{g_m}=S\), (iv) \(a_{2k}(M,g_m)\to+\infty\), \(a_{2k+1}(M,g_m)\to-\infty\), \(k\geq1\), where \(\lambda_k(M,g_m)\) [resp. \(\text{Scal}(g_m)\)] denotes the \(k\)th eigenvalue of the Laplacian \(\Delta_{g_m}\) [resp. the scalar curvature of \((M,g_m)\)].
scientific article; zbMATH DE number 919896