Rigidity theorems for minimal Lagrangian surfaces with Legendrian capillary boundary

TL;DR

This paper establishes rigidity theorems for minimal Lagrangian surfaces with Legendrian capillary boundary in the 4-ball, proving they are either equatorial disks or Lagrangian catenoids, confirming a recent conjecture.

Contribution

It proves that minimal Lagrangian surfaces with Legendrian free boundary are equatorial disks and annulus-type surfaces are Lagrangian catenoids, confirming a conjecture.

Findings

Minimal Lagrangian free boundary surfaces are equatorial disks.

Annulus-type minimal Lagrangian surfaces are Lagrangian catenoids.

Results confirm the conjecture by Li, Wang, and Weng (2020).

Abstract

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).