= Solution
For the axisymmetric <deformation map>, take $\theta=\Theta$, $z=Z$, and $r=r(R)$. A reference cylindrical shell of unit axial length has <volume> $2\pi R\,dR$, and the corresponding current shell has <volume> $2\pi r\,dr$. <Elastic incompressibility> therefore gives $r\,dr=R\,dR$.
Integrating from the inner surface, where $r(A)=a$, gives $r^2-a^2=R^2-A^2$. The positive radius is consequently
$$
\boxed{r(R)=\sqrt{R^2+a^2-A^2},\qquad A\leq R\leq B.}
$$
The outer surface is not an additional independent unknown: $\boxed{b=\sqrt{B^2+a^2-A^2}}$, and hence $h=b-a$.
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