Manifolds of absolutely continuous functions with values in an infinite-dimensional manifold and regularity properties of half-Lie groups
Matthieu F. Pinaud
TL;DR
The paper develops a general framework for spaces of absolutely continuous maps into infinite-dimensional manifolds by using local additions to endow $AC_{L^p}([a,b],N)$ with a smooth manifold structure, and proves smoothness of natural constructions like composition with smooth maps. It then introduces and analyzes $L^p$-semiregularity for right half-Lie groups, establishing the existence and continuity of evolution maps in key examples such as $\mathrm{Diff}_K^r(\mathbb{R}^n)$ and $\mathrm{Diff}^r(M)$ for compact $M$, including $L_{rc}^\infty$ variants and analytic settings. The results provide a robust, general approach to regularity and evolution in infinite-dimensional mapping spaces and half-Lie groups, with concrete implications for the diffeomorphism groups commonly used in geometric analysis and mathematical physics. Overall, the work advances the understanding of how absolute continuity interacts with manifold-valued mappings and how evolution equations can be treated in the setting of infinite-dimensional diffeomorphism groups.
Abstract
For $p\in [1,\infty]$, we define a smooth manifold structure on the set $AC_{L^p}([a,b],N)$ of absolutely continuous functions $γ\colon [a,b]\to N$ with $L^p$-derivatives for all real numbers $a<b$ and each smooth manifold $N$ modeled on a sequentially complete locally convex topological vector space, such that $N$ admits a local addition. Smoothness of natural mappings between spaces of absolutely continuous functions is discussed, like superposition operators $AC_{L^p}([a,b],N_1)\to AC_{L^p}([a,b],N_2)$, $η\mapsto f\circ η$, for a smooth map $f\colon N_1\to N_2$. For $1\leq p <\infty$ and $r\in \mathbb{N}$ we show that the right half-Lie groups $\text{Diff}_K^r(\mathbb{R})$ and $\text{Diff}^r(M)$ are $L^p$-semiregular. Here $K$ is a compact subset of $\mathbb{R}$ and $M$ is a compact smooth manifold. An $L^p$-semiregular half-Lie group $G$ admits an evolution map $\text{Evol}:L^p([0,1],T_e G)\to AC_{L^p}([0,1],G)$, where $e$ is the neutral element of $G$. For the preceding examples, the evolution map $\text{Evol}$ is continuous.