  A geometric $t$-spanner $\mathcal{G}$ on a set $S$ of $n$ point sites in a
metric space $P$ is a subgraph of the complete graph on $S$ such that for every
pair of sites $p,q$ the distance in $\mathcal{G}$ is a most $t$ times the
distance $d(p,q)$ in $P$. We call a connection between two sites in the spanner
a link. In some settings, such as when $P$ is a simple polygon with $m$
vertices and a link is a shortest path in $P$, links can consist of $\Theta
(m)$ segments and thus have non-constant complexity. The total spanner
complexity is a recently-introduced measure of how compact a spanner is. In
this paper, we study what happens if we are allowed to introduce $k$ Steiner
points to reduce the spanner complexity. We study such Steiner spanners in
simple polygons, polygonal domains, and edge-weighted trees.
  Surprisingly, we show that Steiner points have only limited utility. For a
spanner that uses $k$ Steiner points, we provide an $\Omega(nm/k)$ lower bound
on the worst-case complexity of any $(3-\varepsilon)$-spanner, and an
$\Omega(mn^{1/(t+1)}/k^{1/(t+1)})$ lower bound on the worst-case complexity of
any $(t-\varepsilon)$-spanner, for any constant $\varepsilon\in (0,1)$ and
integer constant $t \geq 2$. These lower bounds hold in all settings.
Additionally, we show NP-hardness for the problem of deciding whether a set of
sites in a polygonal domain admits a $3$-spanner with a given maximum
complexity using $k$ Steiner points.
  On the positive side, for trees we show how to build a $2t$-spanner that uses
$k$ Steiner points and of complexity $O(mn^{1/t}/k^{1/t} + n \log (n/k))$, for
any integer $t \geq 1$. We generalize this result to forests, and apply it to
obtain a $2\sqrt{2}t$-spanner in a simple polygon or a $6t$-spanner in a
polygonal domain, with total complexity $O(mn^{1/t}(\log k)^{1+1/t}/k^{1/t} +
n\log^2 n)$.
