On the nodal count for flat tori
Let \(\Delta\) be the scalar Laplacian on a compact Riemannian manifold \((M,g)\) of dimension \(m\). If \(\lambda\) is an eigenvalue of \(\Delta\), let \(E(\lambda,\Delta)\) be the corresponding eigenspace. If \(0\neq\psi\in E(\lambda,\Delta)\), the nodal count \(nc(\psi)\) is the number of connected components of \(\psi^{-1}(\mathbb{R}-\{0\})\). One has \(nc(\psi)\leq C(M)\lambda^{2/m}\) by a theorem of Courant. Thus \(nc_\Delta(\lambda):=\{nc(\psi):\Delta\psi=\lambda\psi\}\) is a finite subset of \(\mathbb{Z}\). We say \((M_1,g_1)\) is isospectral to \((M_2,g_2)\) if the eigenvalue multiplicities are the same. They are said to have the same nodal count if there is a bijection from the projective space \(PE(\lambda,\Delta_1)\) to \(PE(\lambda,\Delta_2)\) preserving the nodal count. The authors use the 4-parameter family of isospectral tori in dimension 4 of \textit{J. H. Conway} and \textit{N. J. A. Sloan} [Int. Math. Res. Not. 1992, No. 4, 93--96 (1992; Zbl 0770.11022)] to provide examples of non-isometric isospectral pairs with the same nodal count and examples of non-isometric isospectral pairs distinguished by their nodal count.
scientific article; zbMATH DE number 6065317
scientific article