Free-Boundary Problems for Holomorphic Curves in the 6-Sphere

TL;DR

This paper investigates free-boundary problems for holomorphic curves in the 6-sphere, establishing rigidity and stability results using complex geometric methods in nearly-Kähler manifolds.

Contribution

It introduces new rigidity and stability results for holomorphic curves with free boundaries in nearly-Kähler 6-manifolds, extending previous geometric analysis techniques.

Findings

Holomorphic curves in geodesic balls meeting boundary orthogonally are totally geodesic.

Rigidity results for reflection-invariant holomorphic curves in S^6.

A topological lower bound on the Morse index for holomorphic curves with Lagrangian boundary conditions.

Abstract

We remark on two different free-boundary problems for holomorphic curves in nearly-K\"{a}hler 6-manifolds. First, we observe that a holomorphic curve in a geodesic ball $B$ of the round 6-sphere that meets $\partial B$ orthogonally must be totally geodesic. Consequently, we obtain rigidity results for reflection-invariant holomorphic curves in $\mathbb{S}^6$ and associative cones in $\mathbb{R}^7$. Second, we consider holomorphic curves with boundary on a Lagrangian submanifold in a strict nearly-K\"{a}hler 6-manifold. By deriving a suitable second variation formula for area, we observe a topological lower bound on the Morse index. In both settings, our methods are complex-geometric, closely following arguments of Fraser-Schoen and Chen-Fraser.