2
$\begingroup$

I am wondering whether there are estimates for the Euclidean length of vertical hyperbolic geodesics for annuli with good geometry.

More precisely: Take an annulus $A$, whose outer boundary $\gamma_{ext} $ is the circle $|z|=1+\epsilon$ and the inner boundary $\gamma_{int}$ is an analytic curve of length at most $M$ (we could set explicitly $M=4\pi$, and take it to be contained in a $2\epsilon$ neighborhood of the unit circle). Let $F$ be the family of vertical hyperbolic geodesics, that is, map $A$ to a straight annulus via the Riemann map and consider the family of curves connecting inner and outer boundary of $A$ defined as the preimages of radial geodesics in the straight annulus. Question: Can we say that there exists K depending on $M, \epsilon$ such that the Euclidean length of every curve in $F$ is at most $K$, regardless of the specific annulus $A$ under consideration?

Comments: 1- you need some condition on the inner boundary, because otherwise you could have arbitrarily long gulfs winding around forcing any vertical geodesics to go through them and hence getting arbitrarily long. I think the condition I stated should be enough-but is it? 2-This type of questions seems to call for the usage of extremal length. But in that way I was only able to get a bound from below on the Euclidean length of the vertical geodesics (in terms of the modulus of $A$). 3-The intuitive idea is that hyperbolic geodesics should follow 'the shortest path', and hence their length should be at most the length of a gulf (which is approx. $\pi$) plus a little extra piece to cross which is say at most $3\epsilon$. How (if it is true) to make this precise?

$\endgroup$
2
$\begingroup$

Yes, you can say things like this.

The easiest way to get these kind of results is probably the Gehring-Hayman theorem. It states that, for two points $z$ and $w$ in a simply-connected domain $D$ or its closure, if there is a curve in $D$ of Euclidean length at most $\ell$ connecting $z$ and $w$, then the hyperbolic geodesic connecting them has length at most $K\cdot \ell$. Here $K$ is a universal constant.

Of course, your domain is not simply-connected, but just lift it under the exponential map and apply the theorem.

Of course, when you have more precise information, you may be able to make better estimates. Harmonic measure often plays an important role in these considerations.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.