Boundary harmonic coordinates on manifolds with boundary in low regularity

TL;DR

This paper establishes the existence of low-regularity harmonic coordinates on 3-manifolds with boundary under minimal curvature and boundary bounds, extending Cheeger-Gromov theory to low regularity boundary settings.

Contribution

It introduces boundary harmonic coordinates for manifolds with boundary under low regularity assumptions, extending geometric analysis techniques to this setting.

Findings

Existence of $H^2$-regular boundary harmonic coordinates under low regularity bounds.

Extension of Cheeger-Gromov convergence theory to manifolds with boundary.

Higher regularity estimates under stronger curvature assumptions.

Abstract

In this paper, we prove the existence of $H^2$-regular coordinates on Riemannian $3$-manifolds with boundary, assuming only $L^2$-bounds on the Ricci curvature, $L^4$-bounds on the second fundamental form of the boundary, and a positive lower bound on the volume radius. The proof follows by extending the theory of Cheeger-Gromov convergence to include manifolds with boundary in the above low regularity setting. The main tools are boundary harmonic coordinates together with elliptic estimates and a geometric trace estimate, and a rigidity argument using manifold doubling. Assuming higher regularity of the Ricci curvature, we also prove corresponding higher regularity estimates for the coordinates.