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Consider a cylinder $C_R$ given by the quotient of the strip $\{z\in \mathbb C|0< Re(z)< R\}$ by the $\mathbb Z$ translation action generated $z\mapsto z+i$, which is endowed with the natural flat metric. It is a classical fact that different $C_R$ can not be conformally equivalent, and there are many different proofs of this fact.

What I wonder is whether there is a more quantitative statement that distinguishes between different $C_R$'s. For example, given $r<R$ consider all possible holomorphic embeddings of $C_r$ into $C_R$, what can we say about the size of the images? For example, is there an estimate on the maximum possible area of the image (with respect to the flat metric)? If we fix $r=1$ and let $R\rightarrow\infty$, how does this maximum area grow in terms of $R$?

I have tried some crude estimate using hyperbolic metrics on $C_R$ and the Schwarz-Ahlfors-Pick lemma. The result I obtained does not seem optimal however.

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There is no estimate of the area of the image. I suppose you consider non-trivial embeddings (inducing non-trivial homomorphisms of the fundamental group). On your cylinder $C_R$ make a cut $[\delta,R]$, where $\delta\in(0,R)$. The resulting region is a ring, and it is conformally equivalent to $C_{r}$ for some $r=r(\delta)$. This $r(\delta)$ is a continuous function, and we have $r(\delta)\to 0$ as $\delta\to 0,$ and $r(\delta)\to R$ as $\delta\to R$. Therefore for every $r\in(0,R)$ there is a non-trivial conformal embedding of $C_r$ into $C_R$. While the area of the image is equal to the area of $C_R$.

That there is also no estimate for trivial embeddings, is evident.

On conformal moduli read Ahlfors, Conformal invariants, McGrow Hill 1973. There is a recent reprint by AMS.

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