A differentiable sphere theorem for compact Lagrangian submanifolds in complex Euclidean space and complex projective space (Q2252675)

The main result of the paper under review is that if \(n\geq 3\) and \(M\) is an \(n\)-dimensional compact Lagrangian submanifold of a complex space form \(\bar M(4c)\) with constant holomorphic sectional curvature \(4c\geq0\), and if the norm square \(S\) of the second fundamental form and the mean curvature \(H\) satisfy \[ S\leq\frac{3n^2H^2}{n+\frac{3}{2}}+2c \] then \(M\) is diffeomorphic to a spherical space form. In the special case when \(M\) is also simply connected, it follows that \(M\) is diffeomorphic to the unit sphere \({\mathbb S}^n\).