the patched Banach $G(F^+\otimes_{{\mathbb{Q}}}{\mathbb{Q}}_p)$-representation $\Pi_\infty$ of [@CEGGPS]. As usual with the patching philosophy, the space $X_p({\overline{\rho}}_x)$ can be related to another geometric object which has a much more local flavour, namely the space $X_{\rm tri}^\square(\overline \rho_{x,p}):=\prod_{v\in S_p}X_{\rm tri}^\square(\overline \rho_{x,v})$ of trianguline representations. More precisely, by [@BHS Th.3.20] there is a Zariski closed embedding: $$\label{maininclusion}
X_p(\overline\rho_x)\hookrightarrow {\mathfrak{X}}_{{\overline{\rho}}_x^p}\times {\mathbb{U}}^g \times X_{\rm tri}^\square(\overline \rho_{x,p}),$$ identifying the source with a union of irreducible components of the target. Here ${\mathbb{U}}^g$ is an open polydisc (related to the patching variables) and ${\mathfrak{X}}_{{\overline{\rho}}_x^p}$ is the rigid analytic generic fiber of the framed deformation space of ${\overline{\rho}}_x$ at all the “bad” places prime to $p$. Moreover the Hecke eigenvariety containing $x$ can be embedded into the patched eigenvariety $X_p(\overline\rho_x)$ (see [@BHS Th.4.2]). As previously, we denote by $x_v$ the image of $x$ in $X_{\rm tri}^\square(\overline \rho_{x,v})$ via (\[maininclusion\]).

For $v\in S_p$ let us write ${\bf k}_{v}$ for the set of labelled Hodge-Tate weights of $\rho_{x,v}$, and $R_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ for the quotient defined in [@Kisindef] of the framed deformation ring of $\overline \rho_{x,v}$ parametrizing crystalline deformations of $\overline \rho_{x,v}$ of Hodge-Tate weight ${\bf k}_v$, and ${\mathfrak{X}}_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm -cr}}$ for the rigid space $({\mathrm{Spf}}\, R_{\overline \rho_{x,v}}^{\square,{\bf k}_v{\rm 