Lie groups of real analytic diffeomorphisms are $L^1$-regular

TL;DR

This paper proves that the Lie group of real-analytic diffeomorphisms on a compact real analytic manifold is $L^1$-regular, ensuring smooth dependence of flows on integrable vector fields, with new tools for infinite-dimensional Lie groups.

Contribution

It establishes $L^1$-regularity for the Lie group of real-analytic diffeomorphisms, introducing new results on regularity and analyticity in infinite-dimensional Lie groups.

Findings

The Lie group of real-analytic diffeomorphisms is $L^1$-regular.

Flows depend smoothly on integrable vector fields.

New results on $L^1$-regularity and analyticity of infinite-dimensional Lie groups.

Abstract

Let $M$ be a compact, real analytic manifold and $G$ be the Lie group of all real-analytic diffeomorphisms of $M$, which is modelled on the space ${\mathfrak g}$ of real-analytic vector fields on $M$. We study flows of time-dependent real-analytic vector fields on $M$ which are integrable functions in time, and their dependence on the time-dependent vector field. Notably, we show that the Lie group $G$ is $L^1$-regular in the sense that each $[\gamma]$ in $L^1([0,1],{\mathfrak g})$ has an evolution which is an absolutely continuous $G$-valued function on $[0,1]$ and depends smoothly on $[\gamma]$. As tools for the proof, we develop new results concerning $L^1$-regularity of infinite-dimensional Lie groups, and new results concerning the continuity and complex analyticity of non-linear mappings on locally convex direct limits.