Relative Liouville Rigidity for Lagrangian Self-Similar Submanifolds with Legendrian Boundary

Abstract

We study compact Lagrangian self-similar immersions in the unit ball with Legendrian boundary. The Liouville form naturally defines a relative de Rham class, and we prove that the vanishing of this relative Liouville class forces the immersion to be a diffeomorphism onto an equatorial Lagrangian disk. In particular, this gives rigidity for exact immersions with connected boundary. We also establish a boundary flux identity and a localized boundary unique continuation theorem, yielding rigidity when a nonempty relatively open boundary portion satisfies the free-boundary condition, when the cosine of the contact angle has a fixed sign, or when the Legendrian capillary boundary is connected. In the two-dimensional minimal case, these results recover the smooth disk theorem of Li--Wang--Weng and the free-boundary theorem of Luo--Sun, while extending the underlying rigidity mechanisms to higher dimensions and the self-similar setting. Finally, in every dimension $n\geq 2$, we construct compact embedded exact Lagrangian examples in the minimal, self-shrinking, and self-expanding cases with two Legendrian capillary boundary components, nonzero relative Liouville class, and supplementary contact angles.