Local gauge conditions for ellipticity in conformal geometry

TL;DR

This paper introduces local gauge conditions in conformal geometry that transform key curvature tensors into elliptic operators, enabling new regularity results and characterizations of conformal flatness.

Contribution

It proposes specific gauge conditions fixing harmonic coordinates and metric normalization to achieve ellipticity of important curvature tensors in conformal geometry.

Findings

Curvature tensors become elliptic under the new gauge conditions.

Elliptic regularity results are established for these tensors.

Characterizations of local conformal flatness are provided in low regularity settings.

Abstract

In this article we introduce local gauge conditions under which many curvature tensors appearing in conformal geometry, such as the Weyl, Cotton, Bach, and Fefferman-Graham obstruction tensors, become elliptic operators. The gauge conditions amount to fixing an $n$-harmonic coordinate system and normalizing the determinant of the metric. We also give corresponding elliptic regularity results and characterizations of local conformal flatness in low regularity settings.