2026/06/30 by Francesco Bei, Simone Cecchini
Mathematics · #math.DG #math.GT
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
arxiv created 2026/07/30 · arxiv updated 2026/07/31
We establish sharp scalar-curvature bounds and rigidity consequences of Gromov's exact-lift two-form method. Let ((Mn,g)), (n≥ 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[infM scalg ≤ -(4n)/(n-1)λ0(X).]Equality forces (g) to be Einstein; if (λ0(X)>0), then (X) is real hyperbolic, while if (λ0(X)=0) and (∫Mωn/2≠ 0), then (g) is flat. The proof combines Gromov's twisted (L2)-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.