Consider a compact surface $M$ with genus $\gamma \geq 2$ and fix a positive real number $V$. Is it known whether it is possible to produce a metric $g$ on the surface $M$ such that $(M. g)$ has volume $V$ and the first eigenvalue of the Laplacian $-\Delta_g$ is as high as one wants? If the answer is yes, what could one make the same conclusions if the metric were additionally required to be negatively curved (but of course not constant negative curvature)? This is mainly a reference request. Somehow, from general heuristics, I am very much willing to believe that the answer to the first question is yes, but I am not at all sure about the second one.
Update: Okay, I just found this paper. See in particular Corollary 1 on page 4, which gives very general bounds on the eigenvalues depending only on the conformal class on a compact manifold of dimension $n$. More explicitly, given a compact Riemannian manifold $M$ and a fixed conformal class $[g]$, it says that the $k$-th eigenvalue is bounded above by a constant $c^{(k)}_{[g]}$ depending on the conformal class, and the aforementioned corollary gives lower bounds for this constant $c^{(k)}_{[g]}$. However, it is not at all clear that this constant $c^{(k)}_{[g]}$ remains bounded as we vary the conformal class.
Furthermore, being not familiar with literature on this kind of investigation, there are a few more questions: (1) If we consider the above Corollary on a closed surface and in the particular case $k =1$, does there exist a smooth metric $g$ in every conformal class such that $\lambda_1(g) = c^{(1)}_{[g]}$? (2) What can we say about the curvature properties of such maximizing metrics?
I am hoping that someone sufficiently familiar with the literature can answer this. Thanks, and sorry if I the question is too long.
