First Robin Eigenvalue of the Laplacian on Kähler and Quaternionic Kähler Manifolds

We investigate the first Robin eigenvalue of the Laplacian on Kähler and quaternionic Kähler manifolds. First, we establish Cheng-type eigenvalue comparison theorems for Kähler manifolds under lower bounds on holomorphic sectional curvature and orthogonal Ricci curvature and for quaternionic Kähler manifolds under scalar curvature lower bounds. For compact Kähler and quaternionic Kähler manifolds, our second result derives lower bounds for the first Robin eigenvalue when the Robin parameter α is positive, with corresponding upper bounds when it is negative.