A friend in physics asked this question, and I didn't know the answer.


Are there lower bounds on the first eigenvalue of the Laplacian of a Riemann surface equipped with a metric of constant negative curvature? 



Lower bound on the first eigenvalue of the Laplacian of a Riemann surface with constant negative scalar curvature
I guess you mean constant curvature $=-1$; otherwise you get an example by rescaling.


On a sphere with two handles there are metrics with curvature $\equiv-1$ which look roughly as a long neck attached to two tori. 
Such metric has as small eigenvalue as you want.


On the other hand upper diameter + lower curvature bound (in particular curvature $\equiv-1$) 
imply a lower bound on the eigenvalue. 


