We study the Steklov eigenvalue functionals $\sigma_k (\Sigma, g) L_g (\partial \Sigma)$ on smooth surfaces with non-empty boundary. We prove that, under some natural gap assumptions, these functionals do admit maximal metrics which come with an associated minimal surface with free boundary from $\Sigma$ into some Euclidean ball, generalizing previous results by Fraser and Schoen in [“Sharp eigenvalue bounds and minimal surfaces in the ball,” Invent. Math., 203(3):823–890, 2016].
"Maximizing Steklov eigenvalues on surfaces." J. Differential Geom. 113 (1) 95 - 188, September 2019. https://doi.org/10.4310/jdg/1567216955