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Uniformly elliptic boundary value problems

Matti Lyko

Abstract

We study boundary conditions for elliptic operators on non-compact manifolds with boundary via uniform K-homology, a version of K-homology sensitive to the large-scale geometry of the manifold. To that end, we develop the theory of relative uniform K-homology. We show that boundary conditions for uniformly elliptic differential operators define classes in the relative and non-relative uniform K-homology of the manifold, depending on the assumed regularity of the boundary condition. Moreover, we define and study a relative index map on relative uniform K-homology that combines uniform coarse information on the interior with secondary information on the boundary. As an application, we compute that on a spin manifold with product structure and uniformly positive scalar curvature on the boundary the image of the relative uniform K-homology class of the Dirac operator under this relative index map is closely connected to a uniform version of the higher $ρ$-invariant of the boundary. In particular, a delocalized APS-index theorem of Piazza and Schick is proved in the uniform setting.

Uniformly elliptic boundary value problems

Abstract

We study boundary conditions for elliptic operators on non-compact manifolds with boundary via uniform K-homology, a version of K-homology sensitive to the large-scale geometry of the manifold. To that end, we develop the theory of relative uniform K-homology. We show that boundary conditions for uniformly elliptic differential operators define classes in the relative and non-relative uniform K-homology of the manifold, depending on the assumed regularity of the boundary condition. Moreover, we define and study a relative index map on relative uniform K-homology that combines uniform coarse information on the interior with secondary information on the boundary. As an application, we compute that on a spin manifold with product structure and uniformly positive scalar curvature on the boundary the image of the relative uniform K-homology class of the Dirac operator under this relative index map is closely connected to a uniform version of the higher -invariant of the boundary. In particular, a delocalized APS-index theorem of Piazza and Schick is proved in the uniform setting.
Paper Structure (57 sections, 191 theorems, 514 equations, 1 figure, 1 table)

This paper contains 57 sections, 191 theorems, 514 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Relative uniform K-homology
  3. Uniform K-homology
  4. Relative uniform K-homology
  5. Paschke duality
  6. Uniform covering isometries
  7. Uniform dual algebras
  8. Paschke duality for relative uniform K-homology
  9. Excision and the long exact sequence
  10. Excision
  11. The long exact sequence
  12. The relative Mayer-Vietoris sequence
  13. Suspension and the Kasparov product
  14. Uniform K-homology of graphs
  15. Index maps on relative uniform K-homology
  16. ...and 42 more sections

Key Result

Theorem 1.1

Let $M$ be a closed manifold, $E\to M$ a vector bundle, and $D$ an elliptic differential operator over $M$. Denote by $D_E$ the operator $D$ twisted by $E$. Then, Here $\mathrm{AS}(D,M)$ is a characteristic class universally constructed from $D$ and $M$, and $\mathrm{ch}(E)$ is the Chern character of $E$.

Figures (1)

  • Figure 1: Sketch of a uniform $C^{k,\alpha}$-domain.

Theorems & Definitions (387)

  • Theorem 1.1: Atiyah-Singer index theorem, AS1968
  • Theorem 1.2: Lichnerowicz, Lichnerowicz1963
  • Theorem 1.3: Atiyah-Bott, Palais1965, AtiyahBott1964
  • Theorem 1.4: Atiyah-Patodi-Singer, APS1
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Theorem 1.9: Relative uniform Paschke duality
  • Theorem 1.10
  • ...and 377 more