Extremal Eigenvalues for Conformally Covariant Operators

Riemannian metrics that extremize eigenvalues of conformally covariant operators are known to have a relationship with the existence of solutions of important partial differential equations (PDEs). On compact surfaces with no boundary, the study of such extremal metrics for the Laplace-Beltrami operator has led mathematicians to special examples of minimal surfaces and harmonic maps into spheres. This thesis is devoted to the study of the existence and properties of extremal metrics for other natural and geometrically defined differential operators. Questions about the regularity of extremal metrics, possible obstructions to their existence, and to which PDEs are these associated with are discussed.