Uniformly elliptic boundary value problems

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.

💡 Research Summary

The paper develops a comprehensive framework for studying boundary value problems of uniformly elliptic differential operators on non‑compact manifolds with boundary, using a version of K‑homology that is sensitive to large‑scale geometry, called uniform K‑homology. The author first extends the existing theory of uniform K‑homology to a relative setting, defining relative uniform K‑homology groups for a pair (X,∂X). This construction relies on Paschke duality for uniform algebras, the introduction of uniform covering isometries, and the development of uniform dual algebras. With these tools, the author proves excision, long exact sequences, and a Mayer‑Vietoris sequence for the relative groups, showing that they retain the formal properties familiar from classical K‑homology.

Next, the paper defines index maps on uniform K‑homology by introducing uniform Roe algebras and their relative versions. The relative index map takes K‑theory classes of the relative uniform Roe algebra to K‑theory of a relative uniform structure algebra, thereby encoding both coarse information from the interior of the manifold and secondary information coming from the boundary. The construction is compatible with the Kasparov product and fits into the exact sequences established earlier.

A substantial part of the work is devoted to the analytic foundations needed for uniformly elliptic operators on manifolds of bounded geometry. The author develops Sobolev spaces that are uniformly controlled, proves a uniform Rellich‑Kondrachov compactness theorem, and establishes trace and extension operators for domains with boundary. These results allow the definition of uniformly elliptic operators with finite propagation and the precise formulation of boundary conditions of varying regularity. Regular boundary conditions give rise to absolute uniform K‑homology classes, while less regular conditions produce relative classes.

Using the abstract machinery, the paper shows that any uniformly elliptic operator D defines a Fredholm module (H,ρ,F) with F = D(D²+1)^{-½}. The commutator