Area Variations Under Legendrian Constraint

In any 5-dimensional closed Sasakian manifold, we prove that any minmax operation on the area among Legendrian surfaces is achieved by a continuous conformal Legendrian map from a closed Riemann surface S into N⁵ N 5 equipped with an integer multiplicity bounded in L^∞ L ∞ . Moreover this map, equipped with this multiplicity, satisfies a weak version of the Hamiltonian Minimal Equation. We conjecture that any solution to this equation is a smooth branched Legendrian immersion away from isolated Schoen–Wolfson conical singularities with non-zero Maslov class.