On $U(1)^{n-2}$-Invariant Special Lagrangian $n$-Folds

TL;DR

This paper extends Joyce's framework to construct and analyze families of $U(1)^{n-2}$-invariant special Lagrangian $n$-folds in complex $n$-space, focusing on their geometry, singularities, and explicit examples.

Contribution

It generalizes Joyce's analytic approach from dimension 3 to arbitrary dimensions, analyzing singularities via polynomial invariants and providing explicit solutions.

Findings

Singularities are governed by an associated polynomial.

Explicit affine and perturbative solutions are constructed.

The analysis extends Joyce's potential and Dirichlet problem methods to higher dimensions.

Abstract

This paper develops a construction of families of $ U(1)^{n-2} $-invariant special Lagrangian $ n $-folds in $ \mathbb{C}^{n} $, extending the analytic framework introduced by Joyce ($ n = 3 $) to arbitrary dimension. By reducing the special Lagrangian condition to a quasilinear elliptic system of two-dimensional non-linear Cauchy-Riemann equations, we analyse both the resulting geometry and its degenerations at singular points. We show that the structure and multiplicity of singularities are governed by an associated polynomial arising from the symmetry reduction. Explicit examples are constructed, including affine and perturbative solutions, and are compared with the classical Harvey-Lawson $ U(1)^{n-1} $-invariant submanifolds. We further show that the key elements of Joyce's analysis in the non-singular case, in particular the potential formulation and Dirichlet problem, extend to this higher-dimensional setting, with the proofs unchanged.