Spectral geometry of Kähler hypersurfaces in a complex Grassmann manifold
Let \(G_{r}(\mathbb{C}^{n})\) denote the Grassmann manifold of complex \(r\)-planes in \(\mathbb{C}^{n}\), equipped with the Kähler metric of maximal holomorphic sectional curvature \(c\). Let \(M\) be a connected compact Kähler hypersurface of \(G_{r}(\mathbb{C}^{n})\) and let \(\lambda_{1}\) denote the first positive eigenvalue of the Laplacian on \(M\). The two following inequalities for the first eigenvalue are proved in the paper: 1. For any \(r\), \(1\leq r\leq n-1\), \( \lambda_{1}\leq c( n-\frac{n-2}{r(n-r)-1}) \). Equality holds only and only if \(r=1\) or \(r=n-1\) and \(M\) is congruent to \(\mathbb{C}P^{n-2}\) in \(\mathbb{C}P^{n-1}\). 2. For \(r=2\), the Grassmann manifold \(G_{2}( \mathbb{C}^{n}) \) admits a quaternionic Kähler structure \(\mathfrak{J}\). If \(M\) is a connected compact Kähler hypersurface such that \(\mathfrak{J} T^{\perp}M\subset TM\), then \( \lambda_{1}\leq c(n-\frac{n-1}{2n-5}) \). Equality holds if and only if \(n=4\) and \(M\) is congruent to a totally geodesic \(Q^{3}\) in the complex quadric \(Q^{4}\simeq G_{2}(\mathbb{C}^{4})\).
scientific article; zbMATH DE number 5015938
scientific article