Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantitative partitioned index theorem and noncompact band-width

Peter Hochs, Jinmin Wang

TL;DR

The paper develops a quantitative partitioned manifold index theory for possibly noncompact hypersurfaces and uses it to extend Gromov's band-width inequality to noncompact bands. By localizing indices near a partitioning hypersurface and employing Callias-type operators, it proves a bound $\inf_{x} ext{Sc}_g(x) \,\le\, rac{4\pi^2(n-1)}{n\ell^2}$ under a coarse-equivalence condition and a nonzero boundary index, and provides an excision framework for the index. Central contributions include the formulation of Ind_q and a quantitative PMIT (Theorem 5.1) plus an explicit noncompact band-width inequality (Theorem 6.1), together with a necessity argument (Theorem 6.2). The results yield sharp obstructions to positive scalar curvature in noncompact band settings and extend Roe/Hochs–Kok constructions to a quantitative, localized regime with coarse-geometric control. These advances deepen the link between scalar curvature, coarse index theory, and large-scale geometric constraints on noncompact manifolds.

Abstract

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.

Quantitative partitioned index theorem and noncompact band-width

TL;DR

The paper develops a quantitative partitioned manifold index theory for possibly noncompact hypersurfaces and uses it to extend Gromov's band-width inequality to noncompact bands. By localizing indices near a partitioning hypersurface and employing Callias-type operators, it proves a bound under a coarse-equivalence condition and a nonzero boundary index, and provides an excision framework for the index. Central contributions include the formulation of Ind_q and a quantitative PMIT (Theorem 5.1) plus an explicit noncompact band-width inequality (Theorem 6.1), together with a necessity argument (Theorem 6.2). The results yield sharp obstructions to positive scalar curvature in noncompact band settings and extend Roe/Hochs–Kok constructions to a quantitative, localized regime with coarse-geometric control. These advances deepen the link between scalar curvature, coarse index theory, and large-scale geometric constraints on noncompact manifolds.

Abstract

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.
Paper Structure (20 sections, 24 theorems, 196 equations, 1 figure)

This paper contains 20 sections, 24 theorems, 196 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A band-width inequality for noncompact bands
  3. Uniformly embedded submanifolds
  4. The main result
  5. A partitioned manifold index theorem for possibly noncompact hypersurfaces
  6. Roe algebras and coarse maps
  7. The partitioned manifold index
  8. A partitioned manifold index theorem
  9. A quantitative partitioned manifold index
  10. Estimates for bounded transforms
  11. Construction of the index in odd dimensions
  12. Construction of the index in even dimensions
  13. Basic properties
  14. The quantitative partitioned manifold index theorem
  15. Statement of the result
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let $X$ be a complete Riemannian $\mathop{\mathrm{Spin}}\nolimits$-manifold of dimension $n$, and let $N \subset X$ be a compact hypersurface. Suppose that Suppose that $M$ is a neighbourhood of $N$ of width $\ell$. Then

Figures (1)

  • Figure 1: Band with infinite length, positive scalar curvature, and non-zero index

Theorems & Definitions (60)

  • Conjecture 1.1: Band-width conjecture; Gromov, 2018
  • Theorem 1.2: Zeidler, 2020
  • Theorem 1.3: Band-width inequality for possibly noncompact bands
  • Definition 2.1
  • Theorem 2.2: Normallyhyperbolicinvariantmanifolds
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5
  • Example 2.6
  • Definition 3.1
  • ...and 50 more