The smallest positive eigenvalue and the length of the shortest geodesic I'm confused about some things concerning lengths of geodesics on Riemann surfaces and positive eigenvalues of the Laplacian. Moreover, I'm also interested in the relation between these two.
Let $X$ be a compact (connected) Riemann surface of genus $g\geq 2$.
Since the complex upper half plane $\mathfrak{h}$ is the universal cover of $X$, we have that $X$ inherits the structure of a Riemannian manifold from  $\mathfrak{h}$. The length of the shortest geodesic with respect to the smooth volume form on $X$ induced by the hyperbolic metric, denoted by $\ell_X$, is well-defined in this case. Let $\lambda_X$ be the smallest positive eigenvalue of the Laplace operator on $L^2(X)$.
Question. What's the relation between $\ell_X$ and $\lambda_X$? Is there some kind of correspondance? 
Now, let's suppose that $b_1,\ldots,b_n$ are points in $X$. Then $X$ is the compactification of a quotient $G\backslash \mathfrak{h} = X\backslash \{b_1,\ldots,b_n\}$ by adding the ``cusps'' $b_1,\ldots,b_n$. (Note that $G \backslash \mathfrak{h}$ inherits the structure of Riemannian manifold from $\mathfrak{h}$.) In this case there is no shortest geodesic on $X$ (due to the existence of cusps). Let $\lambda_G$ be the smallest positive eigenvalue of the Laplace operator on $L^2(G\backslash \mathfrak{h}$.
Question. Does $\lambda_X $ equal $\lambda_G$?
 A: You can have a sequence surfaces $X_i$ with arbitrarily short geodesics but $\lambda_{X_i}$ bounded away from zero. The geodesics are non-separating, so when they shrink to length zero, the Cheeger constant stays bounded away from zero, and therefore so does $\lambda$. You may see this explicitly using Fenchel-Nielsen coordinates. 
On the other hand, there exist surfaces $X_i$ with $\lambda_{X_i}\to 0$ but the shortest geodesic is bounded away from zero. Just take a surface whose shortest geodesic is $\geq N$, and take a large cyclic cover to make $\lambda_X$ arbitrarily small.  
As for filling in cusps, $\lambda_X \neq \lambda_G$ (if I understand your statement correctly). With further geometric information, you can find some relation between them. Robert Brooks has investigated this. 
There is some relation between the lengths of geodesics and the eigenvalues of the Laplacian. In principle, Selberg's trace formula can determine one from the other. If there is a very short geodesic, then one obtains many eigenvalues near $\frac14$, because the spectrum is approaching the continuous spectrum of a cusped surface. 
