Regularity of Lie Groups

We solve the regularity problem for Milnor’s infinite dimensional Lie groups in the $C^0$-topological context, and provide necessary and sufficient regularity conditions for the (standard) $C^k$-topological setting. We prove that the evolution map is $C^0$-continuous on its domain $\mathit{\text{iff}}$ the Lie group $G$ is locally $\mu$-convex. We furthermore show that if the evolution map is defined on all smooth curves, then $G$ is Mackey complete. Under the assumption that $G$ is locally $\mu$-convex, we show that each $C^k$-curve for $k\in $\mathbb{N}${\geq 1}\sqcup{$\mathrm{l}\mathrm{i}\mathrm{p}$,\infty}$ is integrable (contained in the domain of the evolution map) $\mathit{\text{iff}}$ $G$ is Mackey complete and $\mathrm{k}$-confined. The latter condition states that each $C^k$-curve in the Lie algebra $\mathfrak{g}$ of $G$ can be uniformly approximated by a special type of sequence that consists of piecewise integrable curves. A similar result is proven for the case $k\equiv 0$; and, we provide several mild conditions that ensure that $G$ is $\mathrm{k}$-confined for each $k\in $\mathbb{N}$\sqcup{$\mathrm{l}\mathrm{i}\mathrm{p}$,\infty}$. We finally discuss the differentiation of parameter-dependent integrals in the (standard) $C^k$-topological context. In particular, we show that if the evolution map is defined and continuous on $C^k([0,1],\mathfrak{g})$ for $k\in $\mathbb{N}$\sqcup{\infty}$,thenitissmooththereon$$\mathit{\text{iff}}$\hspace{1pt}$itisdifferentiableatzero$$\mathit{\text{iff}}$\hspace{1pt}$ $\mathfrak{g}$ is $$ Mackey$\mathrm{/}$integral$$ complete for $k\in $\mathbb{N}${\geq 1}\sqcup{\infty}\hspace{1pt}/\hspace{1pt}k\equiv 0$. This result is obtained by calculating the directional derivatives explicitly, recovering the standard formulas that hold, e.g., in the Banach case.