Now recall [9, Def. 1.1] a complete manifold $(M,g)$ is called $\text{SOB}(p^{\prime})$, if there exist $x_{0}\in M$ and $C\geq 1$ such that

• •

  $B(x_{0},s)\setminus B(x_{0},t)$ is connected for all $s\geq t\geq C$,

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  $\text{Vol}(B(x_{0},s))\leq Cs^{p^{\prime}}$ for all $s\geq C$,

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  $\text{Vol}(B(x,(1-C^{-1})r(x)))\geq C^{-1}r(x)^{p^{\prime}}$ and $Ric(x)\geq-Cr(x)^{-2}$ if $r(x)=dist(x_{0},x)\geq C$.

Hein [9] shows that when $p^{\prime}>2$, then $SOB(p^{\prime})$ is a sufficient conditions for the weighted Sobolev inequality. As observed in [20, section 7], the connectivity of the annulus can be relaxed to the weaker requirement of relative connected annulus:

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  For sufficiently large $D$, any two points $x_{1},x_{2}\in M$ with $d(x_{0},x_{i})=D$ can be joined by a curve of length at most $CD$, lying in the annulus $B(x_{0},CD)\setminus B(x_{0},C^{-1}D)$, for a uniform constant $C>1$.

For our particular metric ansatz $dd^{c}\phi_{glue}$, the assumptions on the volume growth rate of balls are immediate consequences of the our much more refined description of the generalized Calabi ansatz, and the metric behaviour near the Tian-Yau region. The quadratic decay on the Ricci tensor is a consequence of the faster than quadratic decay on the volume form error to all derivatives (cf. section 4.7).

To verify the relative connnected annulus property, notice any point close to the Tian-Yau region can be first connected via a path of length $O(\tilde{\rho})$ contained inside the annulus, to a point in the generic region where $x_{1},x_{2}$ are comparable, and the statement is obvious for two points in the generic region.