Abstract
We show several results comparing sharp eigenvalue bounds for the first Steklov eigenvalue on surfaces under change of the topology. Among others, we obtain strict monotonicity in the genus. Combined with results of the second named author \cite{petrides_2} this implies the existence of free boundary minimal immersions from higher genus surfaces into Euclidean balls. Moreover, we can also give a new proof of a result by Fraser and Schoen that shows monotonicity in the number of boundary components.
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