# Discrete Isoperimetric Problem in the Hyperbolic Plane

Let $S_{g}$ be the closed orientable genus $g$ surface. I'm interested in studying a certain class $\Gamma$ of graphs on $S_{g}$ which fill (the complementary regions are simply connected), and which hyperbolic metrics on $S_{g}$ optimize the total length of a graph in $\Gamma$ (I'd be happy to make this more explicit).

Let $\mu=(\mu_{1},...,\mu_{n})\in \mathbb{N}^{n}$, and define $\rho_{(\mu)}: \mathbb{(R_{>0})}^{n}\rightarrow \mathbb{R}$ to be the "perimeter function" for $\mu$, which takes as input an $n$-tuple of areas $(\alpha_{1},...,\alpha_{n})$ subject to the constraint

$\sum_{i=1}^{n} \alpha_{i}= 2\pi(2g-2)$, (the area of any hyperbolic surface with topological type $S_{g}$)

and outputs the sum of the perimeters of a collection of $n$ regular polygons, such that the $j^{th}$ polygon has $\mu_{j}$ sides and area $\alpha_{j}$.

$\textbf{Question: For which choice(s) of$(\alpha_{1},...,\alpha_{n})$is$\rho_{(\mu)}$minimized? How does this depend on$\mu$?}$

My first guess was that $\rho_{(\mu)}$ would be minimized by setting all but one area equal to $0$; in other words, put all the area into one polygon, with $\max_{j}(\mu_{j})$ number of sides. But after a bit of experimenting, I think I've convinced myself that there exists choices of $\mu$ for which this fails.

I'm mostly interested in the special case where there is a relationship between $n$ (the number of polygons) and $\sum_{j} \mu_{j}$ (the total number of sides). Specifically, $n= \frac{\sum_{j}\mu_{j}}{4}-2g+2$. In this setting, I'd like to show that the naive guess is right; that is, that the perimeter is optimized by a single regular, right-angled $(8g-4)$-gon.

I've tried looking around to see if people have addressed questions like these and I couldn't find much of anything, although I'm not very familiar with this body of literature.

Any help would be greatly appreciated, and thank you for reading.

$\textbf{UPDATE}:$ I thought it might be useful to give an example of a $\mu$ for which the naive guess is false; suppose $\mu=(k,k,k,...,k)$ for some large $k$, and let $f_{k}(x)$ be the perimeter of a regular hyperbolic $k$-gon with area $x$. The fundamental issue is that $f_{k}(x)$ is not concave-down, unlike in Euclidean geometry where the perimeter of a regular $k$-gon is expressible as some square root of the area. Indeed, one can write down an explicit formula for $f_{k}(x)$ using the usual hyperbolic trig formulas, and the second derivative is positive for $x$ sufficiently large (and of course, smaller than the maximum possible area of a regular $k$-gon). In particular, this means that for certain values of $x$, the most efficient way to enclose an area of $x$ using regular $k$-gons is $\textit{not}$ to use a single $k$-gon. At first, I found this somewhat surprising, but then I realized this is obvious if you choose $x$ to be the area of an ideal $k$-gon.

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Did you check literature on minimal carrier graphs on hyperbolic surfaces? – Misha Jul 15 '13 at 3:53
@Misha: I've come across a bit of the literature on carrier graphs, but not in this context. The particular graphs I'm looking at are a union of two simple closed curves. There exists some nice upper bounds on the length of each edge of a minimal carrier graph on a hyperbolic surface (to my knowledge, due to J. Barnard in this paper), but if anything I'd be looking for lower bounds to rule out the possibility that complementary regions can be very small hyperbolic polygons. – Tarik Aougab Jul 15 '13 at 15:05