A Variational Theory for The Area of Legendrian Surfaces

Abstract: We study a new notion of critical point for the area of surfaces under the Legendrian constraint, called parametrized Hamiltonian stationary Legendrian varifolds (PHSLVs). We establish several fundamental properties of these objects, including their sequential compactness and an optimal regularity result, showing that they are smooth immersions away from a locally finite set of branch points and Schoen Wolfson conical singularities. This generalizes in particular the regularity theory of Schoen Wolfson for minimizers to general critical points. This theory can be used to show two new variational results: every minmax operation with the area of (closed, immersed) Legendrian surfaces in a closed Sasakian 5-dimensional manifold is achieved by a Hamiltonian stationary map with this regularity; also, the minimal area in any given exact isotopy class of Legendrian immersions of $S^2$ is realized by such a map. Along the way, we prove an effective monotonicity formula for general two-dimensional stationary varifolds in the Legendrian setting, as well as the closure of integral stationary varifolds among rectifiable ones, in spite of the lack of compactness of the latter.