Eigenvalue estimates of the drifted Cheng–Yau operator on bounded domains in pinched Cartan–Hadamard manifolds

Abstract We establish universal eigenvalue inequalities for a drifted Cheng–Yau operator on bounded domains in pinched Cartan–Hadamard manifolds with Dirichlet boundary conditions. Our approach combines a universal quadratic eigenvalue inequality for operators in divergence form with a Rauch-type comparison theorem and a Bochner-type formula. We evaluate the geometric terms in this quadratic inequality under suitable assumptions on the radial behavior of the coefficient tensor and the drifting function, together with curvature pinching conditions. In the general setting, we establish a Rauch-type comparison theorem for the Cheng–Yau operator and derive two auxiliary estimates from its Bochner-type formula. We then prove integral estimates of independent interest and apply them to obtain the desired eigenvalue inequalities for both the Cheng–Yau operator and the drifted Laplacian, recovering known inequalities for the Laplacian. We also establish a rigidity result for the Cheng–Yau operator on a class of bounded annular domains in pinched Cartan–Hadamard manifolds. In Euclidean space, this result can be applied, for instance, with the potential function of the Gaussian shrinking soliton. The hyperbolic space is treated separately in two settings, one exploiting the conformal relation between its metric and the Euclidean metric, and the other via its warped product structure, reducing the problem to one dimension. We extend the hyperbolic construction of the fundamental gap for the Laplacian to a class of operators in divergence form. Examples are provided to illustrate the scope and applicability of our results.