Hamiltonian stability of certain minimal Lagrangian submanifolds in complex projective spaces
An immersion \(\varphi:L\to (M,\omega)\) from an \(n\)-dimensional smooth manifold to an \(2n\)-dimensional symplectic manifold is called a {Lagrangian immersion} (and \(L\) a {Lagrangian immersed submanifold}) if \(\varphi^{\ast}\omega\) vanishes identically. A compact Lagrangian submanifold immersed in a Kähler manifold is called (i) \(H\)-minimal if it is a critical point of the volume functional with respect to Hamiltonian variations, (ii) Hamiltonian stable if it is \(H\)-minimal and the second variation for the volume is nonnegative for all Hamiltonian variations. In the paper under review the authors study the following problem: {determine all compact Hamiltonian stable minimal Lagrangian submanifolds in \({\mathbb C}P^n\)}. The main result states the following: let \(L\) be an \(n\)-dimensional compact totally real minimal submanifold embedded in \({\mathbb C}P^n\) with parallel second fundamental form in the list \(SU(p)/{\mathbb Z}_p,\;n=p^2-1\); \(SU(p)/SO(p){\mathbb Z}_p,\;n=(p-1)(p+2)/2\); \(SU(2p)/Sp(p){\mathbb Z}_{2p},\;n=(p-1)(2p+1)\); \(E_6/F_4{\mathbb Z}_3,\;n=26\); then \(L\) is a Hamiltonian stable minimal Lagrangian submanifold in \({\mathbb C}P^n\). The paper includes one section on the fundamental results on Hamiltonian stability of minimal Lagrangian submanifolds in Kähler manifolds and one on the classification theory of totally real submanifolds in \({\mathbb C}P^n\). In the last section the authors study Hamiltonian stability of totally geodesic Lagrangian submanifolds in compact Hermitian symmetric spaces of rank greater than one and they give a complete list of the Hamiltonian stability for all irreducible symmetric \(R\)-spaces of non-Hermitian type canonically embedded in compact irreducible Hermitian symmetric space. The article is well written and the results are interesting and easy to follow.
scientific article; zbMATH DE number 2091878
scientific article