On the Compactness of Hamiltonian Stationary Lagrangian Surfaces in K{\"a}hler Surfaces

TL;DR

This paper establishes convergence and compactness results for sequences of Hamiltonian stationary Lagrangian surfaces in Kähler surfaces, providing foundational insights into their geometric behavior and stability.

Contribution

It proves a bubble tree convergence theorem and strong compactness theorems for Hamiltonian stationary Lagrangian tori in specific Kähler surfaces, advancing understanding of their geometric properties.

Findings

Established bubble tree convergence for sequences with bounded area and energy.

Proved strong compactness for Hamiltonian stationary Lagrangian tori in C2 and CP^2.

Provided new tools for analyzing the compactness of Lagrangian surfaces in Kähler geometry.

Abstract

We prove a bubble tree convergence theorem for a sequence of closed Hamiltonian Stationary Lagrangian surfaces with bounded areas and Willmore energies in a complete K{\"a}hler surface. We also prove two strong compactness theorems on the space of Hamiltonian stationary Lagrangian tori in $\mathbb C^2$ and $\mathbb{CP}^2$ respectively.