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Lagrangian surfaces with Legendrian boundary

Mingyan Li, Guofang Wang, Liangjun Weng

TL;DR

The paper introduces a boundary problem for Lagrangian submanifolds with boundary on the unit sphere, defining Legendrian capillary boundary via a constant contact angle $\theta$ and focusing on the case $n=2$. It develops a Lagrangian analogue of classical free/capillary boundary theory by leveraging convex symplectic geometry, concrete examples, and a Joachimsthal-type boundary property. A key contribution is the showpiece result that minimal Lagrangian disks with Legendrian capillary boundary in $\mathbb{B}^4$ must be equatorial plane disks, extending Nitsche-type rigidity to higher codimension and linking to Whitney spheres and Lagrangian catenoids. The work also introduces the conformal Maslov form framework, uses a holomorphic cubic differential, and proposes conjectures guiding future classification and nonexistence results for annulus-type configurations, with potential impact on understanding boundary behavior of Lagrangian submanifolds in convex symplectic domains.

Abstract

In this note, we first introduce a boundary problem for Lagrangian submanifolds, analogous to the problem for free boundary hypersurfaces and capillary hypersurfaces. Then we present several interesting examples of Lagrangian submanifolds 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.

Lagrangian surfaces with Legendrian boundary

TL;DR

The paper introduces a boundary problem for Lagrangian submanifolds with boundary on the unit sphere, defining Legendrian capillary boundary via a constant contact angle and focusing on the case . It develops a Lagrangian analogue of classical free/capillary boundary theory by leveraging convex symplectic geometry, concrete examples, and a Joachimsthal-type boundary property. A key contribution is the showpiece result that minimal Lagrangian disks with Legendrian capillary boundary in must be equatorial plane disks, extending Nitsche-type rigidity to higher codimension and linking to Whitney spheres and Lagrangian catenoids. The work also introduces the conformal Maslov form framework, uses a holomorphic cubic differential, and proposes conjectures guiding future classification and nonexistence results for annulus-type configurations, with potential impact on understanding boundary behavior of Lagrangian submanifolds in convex symplectic domains.

Abstract

In this note, we first introduce a boundary problem for Lagrangian submanifolds, analogous to the problem for free boundary hypersurfaces and capillary hypersurfaces. Then we present several interesting examples of Lagrangian submanifolds 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.

Paper Structure

This paper contains 8 sections, 7 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Lagrangian surfaces with Legendrian capillary boundary
  3. Convex symplectic manifolds
  4. Interesting examples
  5. A Lagrangian Joachimsthal property
  6. Lagrangian submanifolds with conformal Maslov form
  7. Proof of main theorem
  8. Problems

Key Result

Theorem 1.1

Given $D:=\{(x_1,x_2): x_1^2+x_2^2\leq 1\}$. Let $\iota:D^2\to \mathbb{B}^4\subset \mathbb{R}^4$ be a (branched) minimal Lagrangian surface with Legendrian capillary boundary on $\mathbb{S}^3$. Then $\iota(D)$ is an equatorial plane disk.

Theorems & Definitions (24)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • Example 2.1: Plane disk
  • Example 2.2
  • Proposition 2.6
  • ...and 14 more