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Register a new userRigidity theorems for minimal Lagrangian surfaces with Legendrian capillary boundary
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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).
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