In general for annulus $A\subset \mathbb{C}$ if $A_{1},A_{2}....\subset A$ are disjoint annuli inside it, then we have $$mod(A)=\frac{1}{2\pi}\int_{A}\int_{A} \frac{1}{|z|^{2}}dz>\frac{1}{2\pi}\int_{\cup A_{i}}\int_{\cup A_{i}} \frac{1}{|z|^{2}}dz\geq \sum mod(A_{i}).$$

In my case I have

- shrinking annuli $A_{i}$ separating zero from infinity
- that could possibly intersect on some places $Q_{i,j}:=A_{i}\cap A_{j}\neq \varnothing$
- with, hopefully, controlled diameters $diam(Q_{i,j})\leq d_{i,j}$ going to zero as the annuli shrink
- and the sum $\sum_{i=1}^{N} mod(A_{i})>c_{0}N$ grows linearly.

Let $U_{N}$ be an annulus that contains $\bigcup_{i=1}^{N} A_{i}$.

Q: I am hoping to get a growth for $mod(U_{N})$ eg. $$mod(U_{N})\geq c_{1}\sum_{i=1}^{N} mod(A_{i}).$$

Thank you for any suggestions.

One interesting paper in this direction is "The Quasi-Additivity Law in Conformal Geometry". I will state their result in full generality for the interested reader.

Let S stand for a compact Riemann surface with boundary then for compact subset $K\subset S$ let $W(S,K)=\frac{1}{L(S,K)}$ be the extremal width (reciprocal of extremal length) of paths connecting $\partial S$ to $K$ inside $S\setminus K$.

Consider open sets $A_{i}\subset S$ for i=1,..,N , whose closure is a Riemann surface of finite type (not necessarily connected) with smooth boundary.

Let $X:=W(S,\cup_{i=1}^{N} A_{i})$, $Y:=\sum_{i=1}^{N}W(S,A_{i})$ and $Z:=\sum_{i=1}^{N}W(S\setminus \cup_{j\neq i}^{N} A_{j},A_{i})$.

If $Y<\xi Z$ for some $\xi\geq 1$, there exists constant $K$ depending on $\xi$ and the betti numbers of $S\setminus \cup_{j\in M} A_{j}$ for $M\subset \{1,...,N\}$, such that $$Y\geq K\Rightarrow Y\leq 2\xi X.$$