Estimativas de autovalores para o operador de Cheng-Yau deformado sobre domínios limitados em variedades Cartan-Hadamard pinçadas

Abstract

In this thesis, we show how a Bochner type formula can be used to establish universal inequalities for the eigenvalues of the special case drifted Cheng-Yau operator on a bounded domain in a pinched Cartan-Hadamard manifold with the Dirichlet boundary condition. In the first theorem, the hyperbolic space case is treated in an independent way. For the more general setting, we first establish a Rauch comparison theorem for the Cheng-Yau operator and two estimates associated with the Bochner type formula for this operator. Next, we get some integral estimates of independent interest. As an application, we compute our universal inequalities. In particular, we obtain the corresponding inequalities for both Cheng-Yau operator and drifted Laplacian cases, and we recover the known inequalities for the Laplacian case. We also obtain a rigidity result for a Cheng-Yau operator on a class of bounded annular domains in a pinched Cartan-Hadamard manifold. In particular, we can use, e.g., the potential function of the Gaussian shrinking soliton to obtain such a rigidity for the Euclidean space case. Moreover, we compute the fundamental gap of a class of second-order elliptical differential operators defined on a family of convex domains in the hyperbolic plane.