Fraser–Li first-eigenvalue conjecture for free-boundary minimal hypersurfaces

Determine whether, for every compact properly embedded free-boundary minimal hypersurface of the Euclidean unit ball, the coordinate functions are first Steklov eigenfunctions and the first nonzero Steklov eigenvalue satisfies \(\sigma_1=1\).

Background

The paper recalls the established fact that the coordinate functions of a free-boundary minimal surface are Steklov eigenfunctions with eigenvalue 1. It then distinguishes this fact from the stronger assertion that they are first eigenfunctions. The latter assertion is identified as the Fraser–Li conjecture for compact properly embedded minimal hypersurfaces of the ball. The paper notes that the conjecture has been verified for model examples and, for embedded genus-zero surfaces in $\mathbb B^3$, by Chodosh and Gianocca, but it remains stated in the cited generality as a conjecture.