Lagrangian surfaces with Legendrian boundary

Abstract

In this note, we first introduce a boundary problem for Lagrangian submanifolds, analogous to the free boundary hypersurfaces and capillary hypersurfaces problem. Then we present some interesting minimal Lagrangian submanifolds examples satisfying this boundary condition and we prove a Lagrangian version of Nitsche (or Hopf) type theorem. Some problems are proposed at the end of this note.This project is partly supported by SPP 2026 of DFG "Geometry at infinity".

Free-Boundary Minimal Surfaces of Constant Kahler Angle in Complex Space Forms

“…After this work was completed, I learned of the paper [9] of Mingyang Li, Guofang Wang, and Liangjun Weng, which appeared in print earlier this year. There is some overlap in the Theorem 1.1 of [9] and our Theorem 1.1(a), although neither result implies the other. Moreover, both works deduce the result by showing the vanishing of P .…”

“…In [9], the authors show that a Lagrangian minimal disk in a geodesic ball of M 4 = C 2 that satisfies a Legendrian boundary condition is an equatorial disk. Since the Legendrian boundary condition is more general than the free-boundary condition, this implies our Theorem 1.1(a) in the special case of M 4 = C 2 and Σ homeomorphic to a disk.…”

Rigidity theorems for minimal Lagrangian surfaces with Legendrian capillary boundary

“…It is well known that Lagrangian submanifolds in a complex space form have many similarities with hypersurfaces in a real space form. Recently, inspired by the study of capillary hypersurfaces M in B n+1 ⊂ R n+1 , which have constant mean curvature, non-empty boundary such that M ⊂ Bn+1 and ∂M ⊂ ∂B n+1 = S n , which intersect ∂B n+1 with a constant angle, Li, Wang and Weng [11] initiated the very interesting study of Lagrangian submanifolds with Legendrian capillary boundary in B 2n ⊂ C n .…”

“…First let us recall some definitions introduced in [11]. Let x : Σ n → B 2n be a Lagrangian submanifold with ∂Σ n ⊂ ∂B 2n = S 2n−1 being a Legendrian submanifold.…”

“…When n = 2, typical examples of minimal Lagrangian surfaces in B 4 with Legendrian capillary boundary are the equatorial plane disk and the Lagrangian catenoids, as discussed in [11] (see also Example 3.1). Note that the contact angle for the equatorial plane disk is π 2 , but the contact angle for Lagrangian catenoids are constants which are not equal to π 2 .…”