Uniformly elliptic boundary value problems
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912929431945216 |
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| author | Lyko, Matti |
| author_facet | Lyko, Matti |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_22748 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniformly elliptic boundary value problems Lyko, Matti K-Theory and Homology Differential Geometry 19K56 (primary), 58J32 (secondary) 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. |
| title | Uniformly elliptic boundary value problems |
| topic | K-Theory and Homology Differential Geometry 19K56 (primary), 58J32 (secondary) |
| url | https://arxiv.org/abs/2602.22748 |