Proposition 18.24.2.C8EE003DDDC9 For $u,v\in \mathcal M(\beta )$,
\[ d_\sigma (J(u),J(v))=d\, d_\beta (u,v),\qquad U_\sigma (J(u))=d\, U_\beta (u). \]
The image $J(\mathcal M(\beta ))$ is closed and contains the linear polar-measure geodesic between each pair of its members. The map
\[ R(q)=J\! \left(p_\beta \! \left(d^{-1}r_*\mu _ q\right)\right) \]
is a $1$-Lipschitz retraction onto this image. Its polar measure is
\[ \mu _{R(q)}=d^{-1}r^*r_*\mu _ q. \]
It preserves the polar mass and the non-pluripolar volume, and commutes with ideal projection on $\mathcal M(\sigma )$. At $A=0$ both model spaces are smooth singletons.
Proof.
Change of variables off the finite branch sets gives $\int _{C'}\sigma =dA$. By 5.43.16, the polar measure of $J(v)$ is $r^*\mu _ v$, with total mass $d\mu _ v(C)$. The curve classification gives
\[ \begin{aligned} d_\sigma (J(u),J(v)) & =\| r^*(\mu _ u-\mu _ v)\| _{\mathrm{TV}} =d\| \mu _ u-\mu _ v\| _{\mathrm{TV}},\\ U_\sigma (J(v)) & =dA-d\mu _ v(C)=d\, U_\beta (v). \end{aligned} \]
Completeness of the base model space proves closedness of the image: a convergent sequence in the image comes from a Cauchy sequence downstairs. Linear interpolation of polar measures commutes with $r^*$, giving the stated geodesics.
For any $q\in \mathcal M(\sigma )$, its polar measure has mass at most $dA$. Its pushforward is polar by the image assertion of 5.43.16, so $d^{-1}r_*\mu _ q$ has mass at most $A$ and defines a base model. This proves that $R$ is well defined and gives its displayed polar measure. The identity $r_*r^*=d$ makes $R\circ J=J$. For any finite signed measure $\tau $ on $C'$, pushforward does not increase total variation, hence
\[ \| d^{-1}r^*r_*\tau \| _{\mathrm{TV}} =\| r_*\tau \| _{\mathrm{TV}} \leq \| \tau \| _{\mathrm{TV}}. \]
Apply this to the difference of two polar measures to obtain the Lipschitz estimate. The averaging operator preserves their total mass, so the formula $U_\sigma (q)=dA-\mu _ q(C')$ gives volume preservation. Both measure operations commute with taking the atomic part, which is ideal projection by the curve classification. This proves the commutation assertion. At zero area all polar measures vanish and both spaces are singletons.
$\square$
Pome
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