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 $\lambda_G$?