We establish lower and upper eigenvalue estimates for Dirac operators in different settings, a new Kirchberg type estimate for the first eigenvalue of the Dirac operator on a compact Kähler spin manifold in terms of the energy momentum tensor, and an upper bound for the smallest eigenvalues of the twisted Dirac operator on Legendrian submanifolds of Sasakian manifolds. The sharpness of those estimates is also discussed.
We study the relationship between linear isoperimetric inequalities and the existence of non-constant positive harmonic functions on Riemann surfaces.We also study the relationship between growth conditions of length of spheres and the existence and the existence of Green's function on Riemann surfaces.
An integrated approach to Lie derivatives of spinors, spinor connections and the gravitational field is presented, in the context of a previously proposed, partly original formulation of a theory of Einstein-Cartan-Maxwell-Dirac fields based on “minimal geometric data”: the needed underlying structure is determined, via geometric constructions, from the unique assumption of a complex vector bundle with 2-dimensional fibers, called a -spinor bundle. Any further considered object is assumed to...