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

الأصل · EN

In this note we show the following result using the integral-geometric formula of R. Howard: Consider the totally geodesic RP²ᵐ in CPⁿ. Then it minimizes volume among the isotropic submanifolds in the same Z/2 homology class in CPⁿ (but not among all submanifolds in this Z/2 homology class). Also the totally geodesic RP²ᵐ⁻¹ minimizes volume in its Hamiltonian deformation class in CPⁿ. As a corollary we'll give estimates for volumes of Lagrangian submanifolds in complete intersections in CPⁿ.