Geometric Rigidity via Almost-Harmonic Twisted Spinors

Abstract

We establish sharp scalar-curvature bounds and rigidity consequences of Gromov's exact-lift two-form method. Let ((M^n,g)), (n\geq 4) even, be a closed spin Riemannian manifold carrying a homologically (\widehat A)-non-singular closed two-form (ω) whose lift to the universal cover (X) is exact. Then[\inf_M \operatorname{scal}_g \leq -\frac{4n}{n-1}λ_0(X).]Equality forces (g) to be Einstein; if (λ_0(X)>0), then (X) is real hyperbolic, while if (λ_0(X)=0) and (\int_Mω^{n/2}\neq 0), then (g) is flat. The proof combines Gromov's twisted (L^2)-index with a conformal interpretation of the refined Kato equality and a recentering argument. The same method yields untwisted rigidity results when zero belongs to the spectrum of the Dirac operator on the universal cover, with applications to nonvanishing (\widehat A)-genus and enlargeability.

Substantially revised version. The title has been modified slightly; proofs and exposition have been expanded, with new untwisted rigidity results and applications. Comments are welcome