Two-dimensional Lagrangian singularities and bifurcations of gradient lines II

Abstract

Motivated by mirror symmetry, we consider the Lagrangian fibration ${\mathbb{R}}^4\to {\mathbb{R}}^2$ and Lagrangian maps $f:L\hookrightarrow {\mathbb{R}}^4\to {\mathbb{R}}^2$, exhibiting an unstable singularity, and study how the bifurcation locus of gradient lines, the integral curves of $\nabla f_x$, for $x\in B$, where $f_x(y)=f(y)-x\cdot y$, changes when $f$ is slightly perturbed. We consider the cases when $f$ is the germ of a fold, of a cusp and, particularly, of an elliptic umbilic.

26 pages