Rigidity theorems for minimal Lagrangian surfaces with Legendrian capillary boundary

Abstract

In this note, we study minimal Lagrangian surfaces in B 4 with Legendrian capillary boundary on S 3 . On the one hand, we prove that any minimal Lagrangian surface in B 4 with Legendrian free boundary on S 3 must be an equatorial plane disk. One the other hand, we show that any annulus type minimal Lagrangian surface in B 4 with Legendrian capillary boundary on S 3 must be congruent to one of the Lagrangian catenoids. These results confirm the conjecture proposed by Li, Wang and Weng (Sci. China Math., 2020).