Assume that $S \subset \mathbb{R}^3$ is a smooth, compact, properly embedded surface with boundary $\partial S = \gamma_1 \cup \gamma_2$, where $\gamma_i$ are smooth closed curves. Moreover assume that $S$ is diffeomorphic to an annulus.
If I know the value of the first Dirichlet eigenvalue of $S$, what kind of geometric information do I get? Can I have an estimate on the area, for example?
Any help would be very much appreciated!