Polyhomog\'en\'eit\'e des m\'etriques asymptotiquement hyperboliques complexes le long du flot de Ricci

TL;DR

This paper proves that the polyhomogeneity of asymptotically complex hyperbolic metrics is maintained under Ricci flows, with enhanced results for smooth and Kähler initial metrics, ensuring regularity preservation.

Contribution

It demonstrates the preservation of polyhomogeneity under Ricci-DeTurck and Ricci flows for asymptotically complex hyperbolic metrics, including cases with smooth and Kähler structures.

Findings

Polyhomogeneity is preserved along Ricci-DeTurck flow.

Smoothness up to the boundary is maintained during the flow.

Enhanced results are obtained for Kähler initial metrics.

Abstract

We show that the polyhomogeneity at infinity of an asymptotically complex hyperbolic metric is preserved along the Ricci-DeTurck flow. Moreover, if the initial metric is `smooth up to the boundary', this will be preserved by the Ricci-DeTurck flow and the normalized Ricci flow. When the initial metric is K\"ahler, sharper results are obtained in terms of a potential.