Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $M$ be a geometrically finite hyperbolic surface with one cuspidal end and one funnel end so that it can be divided into $C \cup K \cup F$ where $C$ is the cusp, $F$ the funnel and $K$ the compact core.

Focus on the cusp with the following boundary defining function (due to David Borthwick: Spectral Theory of Infinite-Area Hyperbolic Surfaces)

Let $\rho (x)=e^{-r}$ where $r$ is the distance of $x$ to the compact core $K$. As a boundary defining function it needs to have a non-vanishing differential on the boundary. How can this fact be seen for this function?

If I take the look back to the upper half plane the hyperbolic distance for distinct points $a$ and $b$ located on the y-axis is given by $log(a/b)$. I suppose that's the reason for choosing $e^{-r}$. If you take a point $c$ at the boundary its distance to the compact core is infinite, so $e^{-r(c)}$ is zero, clearly. But what's with the differential? Is there a direct way of calculating it in this case with the given information? I don't understand why is it not zero at the boundary.

In a discussion with others there was the idea to show it by relating the the boundary defining function for the cusp to a (maybe given) boundary defining function for a funnel end by 'inverting', precisely:

Here $M = (0, \infty) \times S^1$ with the metric $g=(dx^2+d\theta^2)/x^2$ where the variable $x$ refers to $(0,\infty)$ and $x$ itself is a boundary defining for the funnel end. If $x$ is bdf for the funnel, is it true that then 1/x is a boundary defining function for the cusp?

My problem is the absence of a diffeomorphism from the cusp (as a point) to the funnel (which is a circle). Can this be repaired by viewing the cusp as a 'infinitely' small circle? The strategy I have in mind is: If there would be a diffeomorphism between the ends and given a boundary defining function for the funnel then the pullback of it has non-vanishing differential.

Since this is my first question on MO, I hope my explanations were precise enough and the questions aren't too basic.

Thank you for your help, Robin Neumann

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.