Convergence of Riemannian manifolds and Albanese tori
Let \(M\) be a compact Riemann manifold and \(H^1(M)_Z\) be a lattice of \(H^1(M)\) which consists of harmonic one-forms of integral periods. Then, by dividing the dual space \(H^1(M)^*\) by the dual lattice \(H^1(M)_Z^*\), one obtains a flat torus, called the Albanese torus of \(M\). Denote by \({\mathcal H} (m, \kappa, D)\) the set of isometry classes of \(m\)-manifolds such that the sectional curvatures are bounded by \(\kappa^2\) and the diameters are not greater than \(D\). Then it is shown here that if \(\{M_i\}\) is a sequence on \({\mathcal H} (m, \kappa, D)\) which converges with respect to the spectral distance (due to the author and \textit{H. Kumura} [Tôhoku Math. J., II. Ser. 46, No. 2, 147-179 (1994; Zbl 0814.53035)], then the Albanese torus \({\mathcal A} (M_i)\) converges to a flat torus \({\mathcal A}_\infty\) of \(\dim n\) with \(0 \leq n \leq \lim \inf_{i \to \infty} b_1(M_i)\), and this proves the convergence of the Albanese tori that collapse to a lower-dimensional space while keeping their curvatures and diameters bounded.
scientific article; zbMATH DE number 871828