## Journal of Differential Geometry

### Maximizing Steklov eigenvalues on surfaces

Romain Petrides

#### Abstract

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].

#### Article information

Source
J. Differential Geom., Volume 113, Number 1 (2019), 95-188.

Dates
First available in Project Euclid: 31 August 2019

https://projecteuclid.org/euclid.jdg/1567216955

Digital Object Identifier
doi:10.4310/jdg/1567216955

Mathematical Reviews number (MathSciNet)
MR3998908

Zentralblatt MATH identifier
07104704

#### Citation

Petrides, Romain. Maximizing Steklov eigenvalues on surfaces. J. Differential Geom. 113 (2019), no. 1, 95--188. doi:10.4310/jdg/1567216955. https://projecteuclid.org/euclid.jdg/1567216955