In this note, we study minimal Lagrangian surfaces in $\mathbb{B}^4$ with Legendrian capillary boundary on $\mathbb{S}^3$. On the one hand, we prove that any minimal Lagrangian surface in $\mathbb{B}^4$ with Legendrian free boundary on $\mathbb{S}^3$ must be an equatorial plane disk. One the other hand, we show that any annulus type minimal Lagrangian surface in $\mathbb{B}^4$ with Legendrian capillary boundary on $\mathbb{S}^3$ must be congruent to one of the Lagrangian catenoids. These results confirm the conjecture proposed by Li, Wang and Weng (Sci. China Math., 2020).
Rigidity of Lagrangian submanifolds with conformal Maslov form and Legendrian capillary boundary
Beijing University of Civil Engineering and Architecture（北京建筑大学）；Chongqing University of Technology（重庆理工大学）；Tsinghua University（清华大学）；Xinyang Normal University（信阳师范大学）
Rigidity theorems for complete Legendrian self-shrinkers
Shanghai Institute for Mathematics and Interdisciplinary Sciences（上海数学与交叉学科研究院）；School of Mathematics and Statistics Wuhan University（武汉大学数学与统计学院）；School of Mathematical Sciences Chongqing University of Technology（重庆理工大学数学科学学院）
Rigidity and Classification of Legendrian Self-Shrinkers
National Taiwan University（台湾大学）；Shanghai Institute of Mathematics and Interdisciplinary Sciences（上海数学与交叉学科研究院）；National Pingtung University（国立屏东大学）；Chongqing University of Technology（重庆理工大学）
