  The conformal Skorokhod embedding problem (CSEP) is a planar variant of the
classical problem where the solution is now a simply connected domain
$D\subset\mathbb{C}$ whose exit time embeds a given probability distribution
$\mu$ by projecting the stopped Brownian motion onto the real axis. In this
paper we explore two new research directions for the CSEP by proving general
bounds on the principal Dirichlet eigenvalue of a solution domain in terms of
the corresponding $\mu$ and by proposing related extremal problems. Moreover,
we give a new and nontrivial example of an extremal domain $\mathbb{U}$ that
attains the lowest possible principal Dirichlet eigenvalue over all domains
solving the CSEP for the uniform distribution on $[-1,1]$. Remarkably, the
boundary of $\mathbb{U}$ is related to the Grim Reaper translating solution to
the curve shortening flow in the plane. The novel tool used in the proof of the
sharp lower bound is a precise relationship between the widths of the
orthogonal projections of a simply connected planar domain and the support of
its harmonic measure that we develop in the paper. The upper bound relies on
spectral bounds for the torsion function which have recently appeared in the
literature.
