such that $\M^{(l)}$ is a smooth manifold of (real) dimension $k\!-\!l$ and $$\ov\M^{(l)}-\M^{(l)}\subset\bigsqcup_{l'=l+1}^{l=k}\!\!\!\M^{(l')}\,.$$ If $U$ is an open subspace of a stratified space $\ov\M$ as above, then $$U=\bigsqcup_{l=0}^{l=k}(\M^{(l)}\!\cap\!U)$$ is also a stratified space. If $\ov\M_1$ and $\ov\M_2$ are stratified spaces, $\ov\M_1\!\times\!\ov\M_2$ is a stratified space with the strata given by unions of the products of the strata of $\ov\M_1$ and $\ov\M_2$. A continuous map $\pi\!:\ov\M_1\!\lra\!\ov\M_2$ between stratified spaces will be called a if the restriction of $\pi$ to each stratum of $\ov\M_1$ is a smooth map to a stratum of $\ov\M_2$. A stratified map $\pi_V\!:V\!\lra\!\ov\M$ will be called a if $\pi_V$ is a topological vector bundle with fiber $\C^k$ and the transition maps from open subsets of $\ov\M$ to $\GL_k\C$ are stratified.\
For the purposes of Definition \[flatfam\_dfn\] below, we set $$\pi_{\std}\!\equiv\!\pi_1: 
\fU_{\std}\equiv\big\{(t,u,v)\!\in\!\C^3\!:~ uv\!=\!t\big\}\lra\C$$ to be the projection to the first component. This is a stratified map with respect to the stratifications $$\C=\C^*\sqcup\{0\}, \qquad 
\fU_{\std}=\pi_{\std}^{-1}(\C^*)\sqcup \big(\pi_{\std}^{-1}(0)\!-\!0\big)
\sqcup\{0\}.$$ For each $t\!\in\!\C^*$, define $$\rho_t\!:\Si_t\!\equiv\!\pi_{\std}^{-1}(t)\lra\R^+ \qquad\hbox{by}\quad 
\rho_t(t,u,v)=u^2+v^2\,.$$ If in addition $\ep\!\in\!\R^+$, let $$\Si_{t,\ep}=\big\{(t,u,v)\!\in\!\Si_t\!:\, |u|^2\!+\!|v|^2<\ep\big\}.$$ If $E\!\lra\!\Si_t$ is a normed vector bundle and $\eta\!\in\!L^p(\Si_t;E)$, let $$\|\eta\|_{t,\ep}=\bigg(\int_{\Si_{t,\ep}}|\eta|^p\bigg)^{1/p}
+\bigg(\int_{\Si_{t,\ep}}\rho_t^{-\frac{p-2}{p}}|\eta|^2\bigg)^{1/2}\,.$$

\[flatfam\_dfn\] A stratified map $\pi\!:\fU\!\lra\!\ov\M$ is a  if

-   each fiber $\Si_u\!\equiv\!\pi^{-1}(u)$ is a (possibly nodal) Riemann 