The Geometric Analysis of Hamiltonian Stationary Surfaces

In the early 1990s, Yong-Geun Oh introduced the problem of studying critical points of the area functional among Lagrangian surfaces in symplectic Riemannian manifolds. Such surfaces are called Hamiltonian stationary or, sometimes, H-minimal surfaces. This variational problem is motivated by natural questions, including the minimization of area within Lagrangian homology classes; within Hamiltonian isotopy classes in Calabi–Yau geometry, as suggested by the Thomas–Yau conjecture; the construction of calibrated minimal surfaces; and even min–max constructions of minimal surfaces in spheres. In the first part of the talk, we will present the significant analytical difficulties associated with the Hamiltonian stationary equation and review the progress made in this direction over the past decades. In the second part, we will study global variational problems concerning the area of surfaces subject to Lagrangian and Legendrian constraints, and address the geometric questions mentioned above.