  Let $(N,g_{0})$ be a Kahler-Einstein surface with the first Chern class
negative and assume that there exists a branched Lagrangian minimal surfaces
with respect to the metric $g_{0}$. We show that when the Kahler-Einstein
metric is changed in the same component (i.e. the complex structure is
changed), the Lagrangian minimal surface can be deformed accordingly. To get
the result, we first obtain a theorem on the deformation of the branched
minimal surfaces in a complete Riemannian $n$-manifold and also generalize a
result of J. Chen and G. Tian on the limit of adjunction numbers.
