16 D. MCDUFF, INTRODUCTION TO SYMPLECTIC TOPOLOGY
Exercise 2.8. Given any two diffeomorphic closed smooth domains U, V in R n
that have the same total volume, show that there is a diffeomorphism ¢ : U , V
that preserves volume. Hint: first choose any diffeomorphism 'ljJ : U , V and look
at the forms Wo,W1 = 'l/J*(wo) on u. Adjust 'ljJ by hand near the boundary au so
that wo = W1 at all points on au. Then use a Moser type argument to make the
forms agree in the interior.
The last important result of this kind is the symplectic isotopy extension the-
orem due to Banyaga. The proof is left as an exercise.
Proposition 2.9 (Isotopy extension). Let Q be a compact submanifold of (M,w)
and suppose that <Pt : M ----> M is a family of diffeomorphisms of M starting at
<Po= id such that ¢;(w) = w near Q. Then, if for every relative cycle Z E H 2 (M, Q)
l ¢;(w) = l w,
there is a family of symplectic diffeomorphisms 'l/Jt and a neighborhood U of Q such
that 'l/Jt(P) = <Pt(P) for all p E U.