In 1982, Yau conjectured that the smallest nonzero eigenvalue of the laplacian on any embedded minimal surface in S^3 is 2; Montiel and Ros later showed that an affirmative answer to Yau's conjecture would imply Lawson's conjecture that the Clifford torus is the only embedded minimal torus in S^3. While the conjecture remains open, Choe-Soret have verified the conclusion for most of the known examples. I will discuss an analogous problem for free boundary minimal surfaces in the unit ball and how this may be used to prove that the critical catenoid is the unique embedded free boundary minimal annulus invariant under reflections through the coordinate planes.