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

Mia S. L. Beard

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.

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

Abstract

This paper develops a construction of families of -invariant special Lagrangian -folds in , extending the analytic framework introduced by Joyce () 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 -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.
Paper Structure (18 sections, 8 theorems, 79 equations, 1 figure)

This paper contains 18 sections, 8 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background on Special Lagrangian Submanifolds
  3. Set-Up of the Geometric Construction
  4. The Symplectic Set-Up
  5. Invariant Coordinates and the Reduced Space
  6. Statement of the Main Result
  7. Proof of the Main Theorem
  8. Tangent Vectors Along the Fibres
  9. Tangent Vectors Transverse to the Group Orbits
  10. Calibrated Cross Product
  11. Proof of Theorem \ref{['THM:MainTheorem']}
  12. Recovery of Joyce's $n = 3$ Construction
  13. Examples
  14. Harvey-Lawson $U(1)^{n-1}$-Invariant Solutions and Translations in $z_{n}$
  15. Affine and Perturbative Special Lagrangians
  16. ...and 3 more sections

Key Result

Proposition 2.4

Let $(\mathbb{C}^{n}, g, \Re(\Omega))$ be equipped with its standard flat metric and calibration. Let $\vec{x} \in \mathbb{C}^{n}$, and let $v_{1}, \dots, v_{n-1} \in T_{\vec{x}} \mathbb{C}^{n}$ be linearly independent vectors spanning an isotropic $(n-1)$-dimensional subspace. Then there exists a u

Figures (1)

  • Figure 1: Visualisation of the relation $P(w) = v^{2} + y^{2}$ and the distinguished solution branch.

Theorems & Definitions (15)

  • Definition 2.1: Special Lagrangian manifolds in $\mathbb{C}^{n}$
  • Definition 2.2: Cross Product on $\mathbb{C}^{n}$
  • Remark 2.3
  • Proposition 2.4: Extension of an $(n-1)$-plane to a special Lagrangian plane
  • Lemma 5.1
  • proof
  • proof : Proof of Theorem \ref{['THM:MainTheorem']}
  • Lemma 7.1: Uniqueness of the reduced triple under a Harvey--Lawson-type gauge fixing
  • proof
  • Lemma 7.2: Existence of $\alpha$
  • ...and 5 more