Compactness of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds (Q6597530)

To a closed connected (possibly immersed) Lagrangian submanifold \(L\) in a Calabi-Yau manifold \(M\) with holomorphic volume form \(\Omega\), one can associate a function \(\theta:L\to\mathbb{R}\) via the relation \(\Omega|_L=e^{i\theta}\mathrm{dvol}_L\). We then call \(L\) special if \(\theta\) is constant. These special Lagrangian submanifolds have been shown to have many interesting properties but are somewhat rare, e.g., there is no closed special Lagrangian submanifold in \(\mathbb{C}^n\). However, inspired by this notion, \textit{Y.-G. Oh} [Math. Z. 212, No. 2, 175--192 (1993; Zbl 0791.53050)] introduced the notion of Hamiltonian stationary Lagrangian submanifolds. These are the (possibly immersed) Lagrangian submanifolds which minimize the volume functional within their Hamiltonian isotopy class. He also showed that, in the Calabi-Yau setting, this is equivalent to being special with \(\theta=0\). These submanifolds are much more common: for example, the standard product torus in \(\mathbb{C}^n\) is Hamiltonian stationary.\N\NIn the present paper, the authors prove the following compactness result for sequences of Hamiltonian stationary Lagrangians \(\{L_k\}\) in a compact symplectic manifold \(M^{2n}\) with Hermitian metric \(h\): If there is some \(C>0\) such that\N\[\N\mathrm{Vol}(L_k)\leq C \quad\text{and}\quad \|A_k\|_{L^n}\leq C,\N\]\Nthen either \(\{L_k\}\) converges to a point, or there is a finite set \(S\) so that a subsequence converges smoothly to a Hamiltonian stationary Lagrangian on \(M\setminus S\). Furthermore, in the latter case, \(\mathrm{Vol}(L)\) is the limit of the volumes given by the subsequence, and the closure of \(L\) is connected, admits the structure of a Lagrangian varifold, and is Hamiltonian stationary in some generalized sense. Here, the volume, the shape operator \(A_k\) of \(L_k\), and the \(L^n\) norms are all computed using the Hermitian metric \(h\). \N\NThis result is a direct generalization of an analogous result by the same authors for Kähler surfaces~[\textit{J. Chen} and \textit{J. M. S. Ma}, Calc. Var. Partial Differ. Equ. 60, No. 2, Paper No. 75, 23 p. (2021; Zbl 1466.53088)] and of the first author with \textit{M. Warren} [J. Differ. Geom. 126, No. 1, 65--97 (2024; Zbl 1542.53081)] for \(\mathbb{C}^n\). In fact, the proof of this new result has very much the same structure as these previous proofs: The idea is always to get estimates on the metric properties of Hamiltonian stationary Lagrangians from the fourth-order elliptic equation defining the Hamiltonian stationary condition by reducing it to simpler systems. In the first paper, this is done by reducing said equation to a coupled system of lower order, while in the second one, this is done by using the very explicit local description of the phase function \(\theta\) in \(\mathbb{C}^n\). In this new paper, the authors instead use a new regularity theory for certain fourth-order equations developed by the first author et al. [Adv. Math. 424, Article ID 109059, 32 p. (2023; Zbl 1515.53079)].\N\NThe paper is structured as follows.\N\begin{itemize}\N\item[1.] In Section~2, the authors discuss some background about Lagrangian submanifolds. They also present some version of the Darboux theorem with precise metric estimates on the chart and the notion of convergence of immersed submanifolds used in the paper.\N\item[2.] In Section~3, the authors use the regularity results for fourth-order equations mentioned above to derive local \(C^k\) estimates on its solutions.\N\item[3.] In Section~4, they prove some regularity results on Hamiltonian stationary Lagrangians using the estimates of the previous section.\N\item[4.] Finally, the proof of the main theorem is split between Sections~5 and~6: 5 for the case \(n=1\) and 6 for the case \(n\geq 2\). This is done so because certain regularity results on the closure of the limit \(L\) are not necessarily true when \(n=1\); however, the proof can then simply be done using classical methods.\N\end{itemize}