Lagrangian surfaces with Legendrian boundary

TL;DR

This paper introduces a boundary problem for Lagrangian submanifolds, provides examples satisfying this condition, and proves a Lagrangian analogue of Nitsche's theorem, contributing to symplectic geometry and boundary value problems.

Contribution

It formulates a new boundary problem for Lagrangian submanifolds, offers explicit examples, and establishes a Lagrangian version of Nitsche's theorem, advancing understanding in symplectic geometry.

Findings

Presented examples of Lagrangian submanifolds with the boundary condition

Proved a Lagrangian version of Nitsche's (or Hopf's) theorem

Proposed open problems for future research

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.