MOSER’S METHOD AND CONSERVATIVE EXTENSIONS OF DIFFEOMORPHISMS

The paper’s Theorem 1 states the problem precisely—if ∂p/∂x1>0 on an open set, there is a local symplectomorphism f with f* p = x1—and proves it by first using the implicit function theorem to arrange p∘g = x1, then applying a Moser homotopy on the symplectic forms ωt to produce a diffeomorphism ρt that preserves x1, finally setting f = ρ1∘g. The key steps (construction of g, exactness ω1−ω0=dµ with µ=(h−x1)dy1, solving a first-order PDE for k via characteristics to get vt, and concluding ρt*ωt=ω0 with x1 fixed) are all present and coherent in the manuscript . The candidate solution instead keeps ω0 fixed and runs a Hamiltonian isotopy φt generated by Ht solving X_{p_t}(Ht)=p−x1 for p_t=(1−t)x1+tp, showing d/dt(p_t∘φt)=0 and hence p∘φ1=x1; local existence of Ht follows from transversality of X_{p_t} to {y1=const}, ensured by ∂p/∂x1>0. This alternative Moser-type approach is standard and correct. Conclusion: both arguments are valid and achieve the same result by different, complementary implementations of Moser’s idea.