For Lawson surfaces with even m and k, the first Laplace eigenvalue equals 2 once a topological obstruction on invariant nodal sets in the fundamental patch is verified.
A new conformal invariant and its applications to the willmore conjecture and the first eigenvalue of compact surfaces.Inventiones Mathematicae, 69(2):269–291, 1982
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Symmetries and the First Laplace Eigenvalue of Lawson Surfaces
For Lawson surfaces with even m and k, the first Laplace eigenvalue equals 2 once a topological obstruction on invariant nodal sets in the fundamental patch is verified.