Locations of Schoen–Wolfson conical singularities

Determine, in general, the possible locations of Schoen–Wolfson conical singularities in Hamiltonian stationary Legendrian surfaces (including area-minimizing representatives in Legendrian homology classes) within Sasakian manifolds, by characterizing where such blow-ups of the form u_{p,q} can occur and under what geometric or topological conditions along the immersed surface.

Background

Schoen–Wolfson established the existence of conical singularities for area-minimizing Hamiltonian stationary Legendrian maps, with blow-ups modeled on explicit cones whose intersections with the unit sphere are (p,q)-torus knots and whose Maslov class reflects the topological nature of the singularity. These cones, called Schoen–Wolfson cones, arise as isolated singular points where the standard regularity theory breaks down.

While the presence and form of such singularities are understood, their precise placement along the surface—i.e., the conditions that dictate where they can appear—has not been fully characterized. Some progress has been made in specific settings, but a comprehensive general description of their possible locations remains lacking.