\section{Miscellaneous}
\label{app:misc}

The following lemma is pretty elementary and rather an intuitive statement on weighted metric spaces.
Obviously, its assumptions are fulfilled by the exponential type weight as chosen in this article.
\begin{lemma}
\label{lem:weighted-M-complete}
    Let some given base weight~$\rho_{\posfold}$ strictly positive and loc\=/Lipschitzian. 
    Denote by~$d_{\riemannmetric*}$ and~$d_{\wriemannmetric*}$ the metrics wrt.\ 
    the nonweighted and weighted Riemannian metric respectively. 
    Then, those metrics induce equivalent topologies. 
    In particular, by~\ref{cond:M:complete} 
    the weighted manifold $(\posfold,\wriemannmetric*)$ is complete as a metric space.
\end{lemma}
\begin{proof}
    Let $(x_n)_{n\in\Nnum}$ be a $d_{\wriemannmetric*}$\=/Cauchy sequence. 
    Since~\posfold is finite dimensional, 
    this sequence is contained in a compactum~$K\subseteq\posfold$. 
    On~$K$ the weight function is continuous, 
    thus it attains minimum and maximum. 
    Let $\varepsilon\in(0,\infty)$. 
    Then, we have for some $N_\varepsilon\in\Nnum$ and all $n_1,n_2\geq N_\varepsilon$ that 
    \begin{displaymath}
        \frac\varepsilon2
        >
        d_{\wriemannmetric*}(x_{n_1},x_{n_2})
        \geq
        \inf_{y\in K} \rho_{\posfold}(y)^2
        \pdot 
        d_{\riemannmetric*}(x_{n_1},x_{n_2})
    .
    \end{displaymath}
    Hence, $(x_n)_{n\in\Nnum}$ is a $d_{\riemannmetric*}$\=/Cauchy sequence 
    and by~\ref{cond:M:complete} it converges to~$x\in\posfold$.
    This $d_{\riemannmetric*}$\=/limit also is the $d_{\wriemannmetric*}$\=/limit:
    \begin{displaymath}
        d_{\wriemannmetric*}(x_{n_1},x)
        \leq
        d_{\wriemannmetric*}(x_{n_1},x_{n_2})
        +
        \norm
            {\rho_{\posfold}^2}
            {\bigL\infty{K}{\Leb{\riemannmetric*}}}
        \pdot 
        d_{\riemannmetric*}(x_{n_2},x)
        < 
        2 \pdot \frac\varepsilon2
        =
        \varepsilon
    \end{displaymath}
    for $n_2\geq N_\varepsilon$ large enough.

    If we start with a $d_{\riemannmetric*}$\=/Cauchy sequence, 
    then it is a $d_{\wriemannmetric*}$\=/Cauchy sequence
    by a similar estimate.
    If the $d_{\wriemannmetric*}$\=/limit exists, 
    it's easily verified that it also is the $d_{\riemannmetric*}$\=/limit.
\end{proof}
{"config": "arxiv", "file": "2006.11567/app-misc.tex"}