First $p$-Steklov eigenvalue under geodesic curvature flow

We study the first nonzero $p$-Steklov eigenvalue on a two-dimensional compact Riemannian manifold with a smooth boundary along the geodesic curvature flow. We prove that the first nonzero $p$-Steklov eigenvalue is nondecreasing if the initial metric has positive geodesic curvature on boundary $\partial M$ and Gaussian curvature is identically equal to zero in $M$ along the un-normalized geodesic curvature flow. An eigenvalue estimation is also obtained along the normalized geodesic curvature flow.