Mass endomorphism and spinorial Yamabe type problems on conformally flat manifolds

摘要

Let M be a compact manifold equipped with a Riemannian metric g and a spin structure sigma. We let lambda(+)(min)(M, [g] sigma) inf ((g) over tilde is an element of[g])lambda(+)(1)((g) over tilde )V ol(M, (g) over tilde)(1) where lambda(+)(1)((g) over tilde) is the smallest positive eigenvalue of the Dirac operator D in the metric (g) over tilde. A previous result stated that lambda(+)(min)(M, [g], sigma) <= lambda(+)(min)(S-n) = n/2 w(n)(1) where w(n) stands for the volume of the standard n-sphere. in this paper, we study this problem for conformally flat manifolds of dimension n >= 2 such that D is invertible. E.g., we show that strict inequality holds in dimension n = 0, 1, 2 mod 4 if a certain endomorphism does not vanish. Because of its tight relations to the ADM mass in General Relativity, the endomorphism will be called mass endomorphism. We apply the strict inequality to spin-conformal spectral theory and show that the smallest positive Dirac eigenvalue attains its infimum inside the enlarged volume-1-conformal class of g.