Morse-Bott Volume Forms

The paper proves a relative Moser theorem for Morse–Bott volume forms via a two-stage construction: a local normal-form/Moser step near Γ using Euler-like vector fields, followed by a global step outside a tubular neighborhood controlled by relative cohomology; the resulting diffeomorphism fixes Γ and pulls back η1 to η0. The candidate solution also succeeds, but via a different one-step global Moser path: it constructs a relative primitive with enhanced vanishing order near Γ using the homotopy operator, defines a time-dependent vector field by contracting with a fixed reference volume form, and integrates it to obtain a diffeomorphism fixing Γ. The model’s write-up is essentially correct but omits a small justification: from [η0]=[η1] in H^n(M,Γ) one must produce a primitive that vanishes (at least) near Γ, which the paper constructs explicitly with a bump-function argument. Once this is added, the model’s approach is fully rigorous.