Willmore Legendrian surfaces in pseudoconformal 5-sphere

Let $X:M\text{hook}{S}^5$ be a compact Legendrian surface in pseudoconformal(CR) 5-sphere. We introduce a pseudoconformally invariant Willmore type second order functional $\text{W}(X)$, and study its critical points called Willmore Legendrian surfaces. The fifth order structure equations show that Willmore dual can be defined for a class of Willmore Legendrian surfaces. Moreover when this dual is constant, Willmore Legendrian surface admits a Weierstra\ss type representation in terms of immersed meromorphic curve in ${\text{C}}^2$ satisfying an appropriate real period condition via pseudoconformal stereographic projection. We show that every compact Riemann surface admits a generally one to one, conformal, Willmore Legendrian immersion in $S^5$ with constant Willmore dual. As a corollary, every compact Riemann surface can be conformally immersed in ${\text{C}}^2$ as an exact, algebraic Lagrangian surface.