  A surface $\Sigma \subset S^5 \subset \mathbb{C}^3$ is called \emph{special
Legendrian} if the cone $0 \times \Sigma \subset \mathbb{C}^3$ is special
Lagrangian. The purpose of this paper is to propose a general method toward
constructing compact special Legendrian surfaces of high genus. It is proved
\emph{there exists a compact, orientable, Hamiltonian stationary Lagrangian
surface of genus $1+\frac{k(k-3)}{2}$ in $ \mathbb{C}P^2$ for each integer $ k
\geq 3$, which is a smooth branched surface except at most finitely many
conical singularities.} If this surface is smooth, it is minimal and the
Legendrian lift of the surface is the desired compact special Legendrian
surface.
  We first establish the existence of a minimizer of area among Lagrangian
disks in a relative homotopy class of a K\"ahler-Einstein surface without
Lagrangian homotopy classes with respect to a configuration $ \Gamma$ that
consists of the fixed point loci of K\"ahler involutions. $ \Gamma$ in addition
must satisfy certain null relative homotopy conditions and angle criteria. The
fundamental domain thus obtained is smooth along the boundary, and has finitely
many interior singular points. We then apply successive \emph{reflection} of
this fundamental domain along its boundary to obtain a complete or compact
Lagrangian surface.
