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 $N^5$ equipped with an integer multiplicity bounded in $L^\infty $. 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.
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Funding
Swiss Federal Institute of Technology Zurich
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