The concentration principle for Dirac operators.

摘要

The symbol map sigma of an elliptic operator carries essential topological and geometrical information about the underlying manifold. We investigate this connection by studying Dirac operators with a perturbation term. These operators have the form Ds = D + sA : Gamma(E) → Gamma( F) over a Riemannian manifold (X, g) for special bundle maps A : E → F and their behavior as s → infinity is interesting. We start with a simple algebraic criterion on the pair (sigma, A) that insures that solutions of Dspsi = 0 localize as s → infinity around the singular set Z A of A. Under certain assumptions of A, ZA is a union of submanifolds, and this gives a new localization formula for the index of