[Paper Review] Constructing special Lagrangian m-folds in C^m by evolving quadrics

Dominic Joyce|ArXiv.org|Aug 21, 2000

Geometric Analysis and Curvature Flows5 references3 citations

TL;DR

This paper constructs explicit special Lagrangian m-folds in ℂ^m by evolving nondegenerate quadrics in ℝ^m via a 1-parameter family of linear or affine maps satisfying a first-order nonlinear ODE. The key contribution is a new method yielding large families of special Lagrangian submanifolds, including cones on S^a × S^b × S^1 for a+b=m−2, with explicit parametrizations for m=3, many of which are new examples of SL T²-cones in ℂ³.

ABSTRACT

This is the second in a series of papers constructing explicit examples of special Lagrangian submanifolds in C^m. The first paper was math.DG/0008021, which studied special Lagrangian m-folds with large symmetry groups. The third is math.DG/0010036, which uses ideas from this paper to construct families of special Lagrangian 3-folds in C^3. This paper describes a construction of special Lagrangian m-folds in C^m which are fibred by (m-1)-submanifolds which are quadrics in Lagrangian planes R^m in C^m. Generically they have only discrete symmetry groups. Some of our examples have been previously constructed by Lawlor and Harvey, using different methods. The principal motivation for these papers is to lay the foundations for the study of singularities of compact special Lagrangian m-folds in Calabi-Yau m-folds. Understanding such singularities will be important in resolving the SYZ conjecture on Mirror Symmetry of Calabi-Yau 3-folds. The special Lagrangian m-folds in C^m we construct here include many cones on S^a x S^b x S^1 for a+b=m-2, which are local models for singularities of special Lagrangian m-folds in Calabi-Yau m-folds.

Motivation & Objective

• To develop a general construction method for special Lagrangian m-folds in ℂ^m using evolving quadrics.
• To provide explicit examples of special Lagrangian submanifolds, particularly cones on products of spheres, as local models for singularities in Calabi–Yau manifolds.
• To extend the known class of special Lagrangian submanifolds beyond those with large global symmetry groups, including new examples in ℂ³.
• To establish conditions under which the evolution flow is periodic, leading to compact or closed special Lagrangian submanifolds.
• To connect the construction to integrable systems via harmonic tori in ℂℙ^m for m=3.

Proposed method

• The construction evolves an (m−1)-dimensional submanifold P in ℝ^n via a 1-parameter family of linear or affine maps φ_t: ℝ^n → ℂ^m.
• The maps φ_t satisfy a first-order nonlinear ordinary differential equation in t derived from the special Lagrangian condition.
• For the main case, P is a nondegenerate quadric in ℝ^m (e.g., ellipsoid, hyperboloid, paraboloid), and the evolution is governed by a system of ODEs on complex-valued functions w₁, w₂, and β.
• The linear version evolves centered quadrics (e.g., ellipsoids) using linear maps; the affine version evolves non-centered quadrics (e.g., paraboloids) using affine maps.
• The resulting submanifold N ⊂ ℂ^m is the image of P under the flow φ_t, and is shown to be special Lagrangian when the ODE system is solved consistently.
• For m=3, the ODEs reduce to solvable systems, yielding explicit parametrizations of special Lagrangian 3-folds in ℂ³, including cones on S^a × S^b × S^1.

Experimental results

Research questions

• RQ1 Can special Lagrangian m-folds in ℂ^m be systematically constructed by evolving quadric surfaces in ℝ^m via a 1-parameter family of linear or affine maps?
• RQ2 Under what conditions does the evolution flow yield closed or periodic submanifolds, particularly cones on products of spheres?
• RQ3 What is the topology and geometry of the resulting special Lagrangian submanifolds, especially in the case m=3?
• RQ4 How do these constructions relate to integrable systems and harmonic tori in complex projective space?
• RQ5 Are there new examples of special Lagrangian 3-folds in ℂ³, particularly SL T²-cones, not previously known?

Key findings

• The paper constructs large families of special Lagrangian m-folds in ℂ^m by evolving quadrics, with the first such construction for non-centered quadrics (e.g., paraboloids) via affine maps.
• For m=3, explicit parametrizations are derived for special Lagrangian 3-folds, including cones on S^a × S^b × S^1 with a+b=1, yielding new examples of SL T²-cones in ℂ³.
• When the evolution flow is periodic, the resulting special Lagrangian m-folds are closed and embedded, with topology ℝ^m or more complex structures depending on initial data.
• The construction yields examples with only finite global symmetry groups, contrasting with earlier constructions based on large global symmetry groups.
• In the case m=3, the solutions to the ODEs are explicitly solved, giving parametric expressions for N in terms of complex parameters C, D, and real parameters x₁, x₂, t.
• The construction recovers known examples (e.g., Lawlor–Harvey cones) but also produces new ones, particularly when Im(C𝒟̄) ≠ 0 or |C| ≠ |D|, leading to non-planar, embedded SL 3-folds in ℂ³.