Estimate for the first fourth Steklov eigenvalue of a minimal hypersurface with free boundary

Abstract

. We explore the fourth-order Steklov problem of a compact embedded hyper-surface (cid:54) n with free boundary in a ( n + 1 ) -dimensional compact manifold M n + 1 which has nonnegative Ricci curvature and strictly convex boundary. If (cid:54) is minimal we establish a lower bound for the first eigenvalue of this problem. When M = B n + 1 is the unit ball in (cid:82) n + 1 , if (cid:54) has constant mean curvature H (cid:54) we prove that the first eigenvalue satisfies σ 1 ≤ n + | H (cid:54) | . In the minimal case ( H (cid:54) = 0), we prove that σ 1 = n .