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

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.

arXiv commentv5: 47 pages; typos removed, minor improvements

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