Volume minimization and estimates for certain isotropic submanifolds in complex projective spaces

In this note we show the following result using the integral-geometric formula of R. Howard: Consider the totally geodesic $\mathbb{R}{P}^{2m}$ in $\mathbb{C}{P}^n$. Then it minimizes volume among the isotropic submanifolds in the same $\mathbb{Z}\mathrm{/}2$ homology class in $\mathbb{C}{P}^n$ (but not among all submanifolds in this $\mathbb{Z}\mathrm{/}2$ homology class). Also the totally geodesic $\mathbb{R}{P}^{2m-1}$ minimizes volume in its Hamiltonian deformation class in $\mathbb{C}{P}^n$. As a corollary we’ll give estimates for volumes of Lagrangian submanifolds in complete intersections in $\mathbb{C}{P}^n$.