turkmath.org

The geodesics for the Mabuchi metric on the space H of Kahler metrics on a compact symplectic manifold M correspond to solutions of a homogeneous complex Monge-Ampere (HCMA) equation. The space H is an infinite dimensional analogue of the symmetric spaces of noncompact type G_C/G for compact Lie groups G. In H the role of G is being played by the group of Hamiltonian symplectomorphisms. I will describe a method for reducing the relevant Cauchy problem for the HCMA eq with analytic initial data to finding a related Hamiltonian flow followed by a "complexification". For Hamiltonian G-spaces, with G-invariant Kahler structure, the geodesic corresponding to the norm square of the moment map or its Hamiltonian flow in imaginary time (= gradient flow for the changing metric following the geodesic) leads to the convergence of the holomorphic sections to sections supported on Bohr-Sommerfeld leaves. For M=T*G, starting from the vertical or Schrodinger polarization, one obtains the Segal-Bargman-Hall coherent state transform.