The singular sets of degenerate and nonlocal elliptic equations on Poincar\'e-Einstein manifolds

TL;DR

This paper investigates the structure and measure estimates of singular sets for degenerate and nonlocal elliptic operators on Poincaré-Einstein manifolds, linking geometric analysis with conformal geometry.

Contribution

It develops a quantitative differentiation framework and new regularity results for these operators, advancing understanding of their singular sets in geometric analysis.

Findings

Quantitative stratification of singular sets

Minkowski type estimates for stratified singular sets

Hausdorff measure bounds for singular sets

Abstract

The main objects of this paper include some degenerate and nonlocal elliptic operators which naturally arise in the conformal invariant theory of Poincar\'e-Einstein manifolds. These operators generally reflect the correspondence between the Riemannian geometry of a complete Poincar\'e-Einstein manifold and the conformal geometry of its associated conformal infinity. In this setting, we develop the quantitative differentiation theory that includes quantitative stratification for the singular set and Minkowski type estimates for the (quantitatively) stratified singular sets. All these, together with a new $\epsilon$-regularity result for degenerate/singular elliptic operators on Poincar\'e-Einstein manifolds, lead to uniform Hausdorff measure estimates for the singular sets. Furthermore, the main results in this paper provide a delicate synergy between the geometry of Poincar\'e-Einstein manifolds and the elliptic theory of associated degenerate elliptic operators.