Evolution of the Steklov eigenvalue under geodesic curvature flow

Abstract

On a two-dimensional compact Riemannian manifold with boundary, we prove that the first nonzero Steklov eigenvalue is nondecreasing along the unnormalized geodesic curvature flow if the initial metric has positive geodesic curvature and vanishing Gaussian curvature. Using the normalized geodesic curvature flow, we also obtain some estimate for the first nonzero Steklov eigenvalue. On the other hand, we prove that the compact soliton of the geodesic curvature flow must be the trivial one.