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@We prove that any U(1)^n-orbit in P^n is Hamiltonian volume minimizing, i.e., volume minimizing under Hamiltonian deformation. The idea is the following :
(1) We extend one U(1)^n-orbit to the moment torus fibration {T_t:tธDelta_n} and consider its Hamiltonian deformation {ำ(T_t):tธDelta_n} where ำ is a Hamiltonian diffeo-morphism of P^n and then :
(2) We compare each U(1)^n-orbit and its Hamiltonian deformation by comparing the large k asymptotic behavior of the sequence of projective embeddings defined, for each k, by the basis of H^0(P^n,mathcal{O}(k))) obtained by semi-clasasical approximation of the mathcal{O}(k)-BS (Bohr-Sommerfeld) tori of the Lagrangian torus fibrations {T_t:tธDelta_n} and its Hamiltonian deformation {ำ(T_t):tธDelta_n}.
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