ИСТИНА

Hamiltonian minimality (H-minimality) for Lagrangian submanifolds is a symplectic analogue of minimality in Riemannian geometry. A Lagrangian immersion is called H-minimal if the variations of its volume along all Hamiltonian vector fields are zero. A family of H-minimal Lagrangian submanifolds N in a complex space C^m can constructed from intersections of real or Hermitian quadrics. These intersection of quadrics are parametrised by convex simple polytopes, appear in the symplectic reduction construction of Hamiltonian toric manifolds, and are known in toric topology under the name moment-angle manifolds. The topology of moment-angle manifolds is complicated, but relatively well understood, and its knowledge can be used to describe new H-minimal Lagrangian submanifolds N with interesting topology. For example, starting from a polygon, one obtains as N a twisted product of a torus and a Riemannian surface of large genus. The construction of H-minimal Lagrangian submanifolds N in C^m can be enhanced by applying it alongside with the symplectic reduction, which lead to new examples of H-minimal submanifolds in a projective space and other toric varieties. Furthermore, manifolds N (both embedded and immersed) appear as degenerations of Liouville tori for some interesting Hamiltonian systems with polynomial integrals. The talk is based on joint works with Andrey E. Mironov.