# Free Boundary Hamiltonian Stationary Lagrangian Discs in ℂ2

## TL;DR

This paper investigates conditions under which certain geometric surfaces in complex space are minimal and provides examples to validate these conditions.

## Contribution

The paper establishes new conditions for Hamiltonian stationary Lagrangian discs to be minimal and demonstrates their optimality with examples.

## Key findings

- Weakly conformal, branched free boundary Hamiltonian stationary Lagrangian discs are minimal under specific conditions.
- Such discs with Legendrian boundary are shown to be Lagrangian equatorial plane discs.
- Examples are provided to confirm the optimality of the derived conditions.

## Abstract

Let \documentclass[12pt]{minimal}
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\begin{document}$$\Omega \subset {\mathbb {C}}^2$$\end{document}Ω⊂C2 be a smooth domain. We establish conditions under which a weakly conformal, branched \documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document}Ω-free boundary Hamiltonian stationary Lagrangian immersion u of a disc in \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {C}}^2$$\end{document}C2 is a \documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document}Ω-free boundary minimal immersion. We deduce that if \documentclass[12pt]{minimal}
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\begin{document}$$u$$\end{document}u is a weakly conformal, branched \documentclass[12pt]{minimal}
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\begin{document}$$B_1(0)$$\end{document}B1(0)-free boundary Hamiltonian stationary Lagrangian immersion of a disc with Legendrian boundary, then \documentclass[12pt]{minimal}
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\begin{document}$$u(D^2)$$\end{document}u(D2) is a Lagrangian equatorial plane disc. Furthermore, we present examples of \documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document}Ω-free boundary Hamiltonian stationary discs, demonstrating the optimality of our assumptions.