Two-dimensional Lagrangian singularities and bifurcations of gradient lines I

Motivated by mirror symmetry, we consider a Lagrangian fibration X→BX→B and Lagrangian maps f:L↪X→Bf:L↪X→B, when L   has dimension 2, exhibiting an unstable singularity, and study how their caustic changes, in a neighbourhood of the unstable singularity, when slightly perturbed. The integral curves of ∇fx∇fx, for x∈Bx∈B, where fx(y)=f(y)−x⋅yfx(y)=f(y)−x⋅y, called “gradient lines”, are then introduced, and a study of them, in order to analyze their bifurcation locus, is carried out.