Mathematics, Differential Geometry, Differential Geometry (math.DG), Analysis of PDEs (math.AP)

journal:

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date:

2023-06-17 16:00:00

Abstract

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.