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

Edward Goldstein
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Abstract

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 $\mathbb{Z}/2$ homology class in $\mathbb{C}P^n$ (but not among all submanifolds in this $\mathbb{Z}/2$ 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 $\mathbb{C}P^n$.

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Edward Goldstein. 2004-12-31. Volume minimization and estimates for certain isotropic submanifolds in complex projective spaces. https://arxiv.org/abs/math/0406334

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