Lagrangian Submanifolds with Legendrian Boundary in the Unit Ball

We study compact Lagrangian submanifolds in the unit ball ${\mathbb{B}}^{2n}\subset {\mathbb{C}}^n$ with Legendrian boundary. We prove that any compact exact Lagrangian self-similar submanifold with connected Legendrian boundary must be an equatorial $n$-disk. The same rigidity holds, without exactness, for Legendrian boundary under a fixed sign assumption on the cosine of the contact angle; in particular, it holds for Legendrian free boundary. These results extend the two dimensional minimal rigidity theorems of Li-Wang-Weng and Luo-Sun to higher dimensions and to the Lagrangian self-similar setting, which includes the minimal case. Notably, the Legendrian capillary condition in Li-Wang-Weng’s theorem is weakened to the Legendrian boundary condition. Our proof uses the Liouville form and boundary unique continuation for differential forms, rather than holomorphic differential techniques. Finally, we construct non-disk-type Lagrangian self-similar examples with Legendrian capillary boundary.