# Modulus of an annulus with a cut

Let $A_r$ be a complex annulus of modulus $r>0$ obtained from a $1\times r$ rectangle in $\mathbb C$ with vertices $A=0$, $B=r$, $C=r+i$, $D=i$, by identifying isomterically $AB$ with $DC$.

Let us now chose $0<s<r$ and make a cut of the cylinder $A_r$ along the segment $[0,s]$ (if $s=r$ we get back the rectangle). Denote the obtained annulus by $A_r(s)$. Clearly, the modulus of $A_r(s)$ is less than $r$.

Question. Is there a reasonably looking estimation from above of the modulus $M(A_r(s))$ of cylinder $A_r(s)$? More precisely I wonder if there is a relatively simply looking continuous function, $f(r,s)$ defined on the domain $0\le s\le r$ such that

$$f(r,s)\ge M((A_r(s)),$$ $$f(r,0)=r,\;\;f(r,s_1)>f(r,s_2)\; if \;\; s_2>s_1,\;\; f(r,r)=0.$$

• That the modulus tends to zero when $s$ tends to $r$ follows from the definition of extremal length. You can also compare with the modulus of the Grötzsch annulus, about which much is known. – Lasse Rempe-Gillen Apr 21 '17 at 12:53
• Of course, as $s$ tends to 0, the modulus tends to the original modulus $r$. To obtain bounds, you could use an explicit quasiconformal map taking your annulus to the original one, and bound its dilatation. More sophisticated techniques can undoubtedly be used, but it depends on your needs whether these would be necessary. – Lasse Rempe-Gillen Apr 21 '17 at 12:56
• One can write an explicit formula for this modulus, in terms of elliptic functions. But do you really need it? Please state what exactly you need to know about this modulus. – Alexandre Eremenko Apr 21 '17 at 13:34
• Lasse, thanks, indeed this is very similar to Grotzsch annuls. Probably I can use it to address my question – aglearner Apr 21 '17 at 14:17