Rigidity theorems for minimal Lagrangian surfaces with Legendrian capillary boundary

Abstract

In this note, we study minimal Lagrangian surfaces in B4 with Legendrian capillary boundary on S3. On the one hand, we prove that any minimal Lagrangian surface in B4 with Legendrian free boundary on S3 must be an equatorial plane disk. On the other hand, we show that any annulus type minimal Lagrangian surface in B4 with Legendrian capillary boundary on S3 must be congruent to one of the Lagrangian catenoids. These results confirm the conjecture proposed by Li, Wang and Weng [12].