Polyhomog\'eni\'et\'e des m\'etriques compatibles avec une structure de Lie \`a l'infini le long du flot de Ricci

TL;DR

This paper investigates how certain complete Riemannian metrics with a fibred Lie structure at infinity maintain their polyhomogeneity under Ricci flow, especially for asymptotically Einstein initial metrics, using advanced geometric analysis techniques.

Contribution

It proves that polyhomogeneity of metrics with a fibred Lie structure at infinity is preserved under Ricci-DeTurck flow, extending previous results to a broader class of metrics.

Findings

Polyhomogeneity is locally preserved by Ricci-DeTurck flow.

Asymptotically Einstein initial metrics retain polyhomogeneity during flow.

Smooth boundary conditions are preserved under normalized Ricci flows.

Abstract

Along the Ricci flow, we study the polyhomogeneity of complete Riemannian metrics endowed with "a Lie structure fibred at infinity", that is, a class of Lie structures at infinity that induce in a precise way a fibre bundle structure on a certain compactification by a manifold with corners. When the compactification is a manifold with boundary, this class of metrics contains, in particular, the b-metrics of Melrose, the fibred boundary metrics of Melrose and Mazzeo and the edge metrics of Mazzeo. Our main result consists in showing that the polyhomogeneity of the metrics compatible with a Lie structure fibered at infinity is locally preserved by the Ricci-DeTurck flow. If the initial metric is asymptotically Einstein, the polyhomogeneity of the metrics solutions is obtained as long as the flow exists. Moreover, if the initial metric is "smooth up to the boundary", then it will be also preserved by the normalized Ricci flow and the Ricci-DeTurck flow.