Quantitative partitioned index theorem and noncompact band-width

Gromov’s band-width conjecture gives a precise upper bound for the width of a compact Riemannian band with positive scalar curvature lower bound, assuming that the cross-section of the band admits no positive scalar curvature metrics. Versions of this were proved by Cecchini and by Zeidler. In this paper, we develop a quantitative version of partitioned manifold index theory, which applies to noncompact hypersurfaces. Using this, we prove a version of Gromov’s band-width estimate for possibly noncompact Riemannian bands.

💡 Research Summary

This paper makes a significant advance in geometric analysis by proving a noncompact version of Gromov’s band-width conjecture. The conjecture posits an optimal upper bound (4π²(n-1)/(nℓ²)) for the width ℓ of a Riemannian band (a manifold with two boundary components) with a positive scalar curvature lower bound, provided the cross-section of the band does not admit positive scalar curvature metrics. While previous proofs by Cecchini and Zeidler required compact cross-sections, this work removes that compactness assumption.

The authors’ primary achievement is the development of a quantitative partitioned manifold index theorem applicable to noncompact hypersurfaces. In index theory, Roe’s partitioned manifold index theorem relates the index of a Dirac operator on a manifold split by a compact hypersurface to the index of a Dirac operator on the hypersurface itself. Generalizations exist for noncompact hypersurfaces, but they typically express the equality in the K-theory of a localized Roe algebra—an inductive limit of algebras associated to larger and larger neighborhoods of the hypersurface. This lacks the quantitative control needed for geometric estimates.

The key innovation here is the construction of a quantitative index, Ind_q(D, N), which lives in the K-theory of the Roe algebra C*(M) of a specific, controlled closed set M containing the hypersurface N. The main theorem (Theorem 5.1) establishes that this quantitative index equals the image of the coarse index of the Dirac operator on N under the map induced by the inclusion i_N: N → M. This “localization” of the index to a predetermined set M is what makes the theory quantitative.

The authors then apply this powerful tool to prove their main geometric result (Theorem 2.5/1.3). It states that for a regular Spin Riemannian band (M,g) of width ℓ, if 1) the intrinsic distance on ∂-M is coarsely equivalent to the distance induced from M, and 2) the coarse index of the Dirac operator on ∂-M is non-zero, then the infimum of the scalar curvature on M satisfies inf Sc_g ≤ 4π²(n-1)/(nℓ²). The first condition is a subtle but crucial geometric control on how the boundary is embedded within the band, and the paper provides examples showing its necessity. The proof proceeds by contradiction: assuming a stricter scalar curvature lower bound allows the construction of an invertible Callias-type operator, which forces the quantitative index (and hence the coarse index of ∂_-M) to vanish, contradicting assumption 2.

In summary, this work successfully bridges a gap between high-level index theory and concrete geometric problems. By refining the partitioned manifold index theory to a quantitative, localized form, the authors extend the reach of the Dirac operator method to analyze the geometry of noncompact bands with positive scalar curvature, yielding a natural generalization of the celebrated band-width estimate.