[Paper Review] Special Lagrangian conifolds, I: Moduli spaces (extended version)

Tommaso Pacini|arXiv (Cornell University)|Nov 12, 2012

Geometry and complex manifolds12 references3 citations

TL;DR

This paper establishes a deformation theory for special Lagrangian (SL) conifolds in $\mathbb{C}^m$, generalizing McLean's theory to include both conically singular (CS) and asymptotically conical (AC) ends. It proves that the moduli space of SL conifolds is a smooth, finite-dimensional manifold whose dimension is determined by the index of a weighted Laplacian, with topological and analytic contributions from CS and AC ends respectively.

ABSTRACT

This is the extended version of the paper "Special Lagrangian conifolds, I: Moduli spaces", which discusses the deformation theory of special Lagrangian (SL) conifolds in complex space C^m. Conifolds are a key ingredient in the compactification problem for moduli spaces of compact SLs in Calabi-Yau manifolds. The conifold category allows for the simultaneous presence of conical singularities and of non-compact, asymptotically conical, ends. Our main theorem is the natural next step in the chain of results initiated by McLean and continued by the author and Joyce. We survey all these results, providing a unified framework for studying the various cases and emphasizing analogies and differences. Compared to "Special Lagrangian conifolds, I", this paper contains more detail but the same results. The paper also lays down the geometric foundations for our paper "Special Lagrangian conifolds, II: Gluing constructions in C^m".

Motivation & Objective

• To develop a unified deformation theory for special Lagrangian conifolds in $\mathbb{C}^m$, encompassing both conically singular and asymptotically conical ends.
• To extend McLean's deformation theory for compact SLs to non-compact, singular SLs, enabling the study of compactification of moduli spaces.
• To provide the geometric and analytic foundation for gluing constructions of SL conifolds, as required in Joyce's compactification program.
• To characterize the dimension of the moduli space of SL conifolds via the index of a weighted Laplacian operator on weighted Sobolev spaces.
• To clarify how decay conditions at AC ends (via compactly supported cohomology) and CS singularities (via vanishing restriction maps) contribute to the moduli space dimension.

Proposed method

• Formalizes the category of Riemannian conifolds as manifolds with both conically singular and asymptotically conical ends.
• Introduces weighted Sobolev spaces $W^{p}_{k,(oldsymbol{ u},oldsymbol{ ho})}$ to handle decay and growth rates at AC and CS ends, respectively.
• Analyzes the linearized SL deformation operator via the weighted Laplacian $\Delta_{\boldsymbol{\mu},\boldsymbol{\lambda}}$ on these weighted spaces.
• Applies the implicit function theorem to the nonlinear deformation map $\tilde{F}$, proving that its zero set is a smooth manifold when the linearized operator is an isomorphism.
• Uses the change of index formula to relate the dimension of the moduli space to the difference in indices of the Laplacian at different weights.
• Relies on the isomorphism of the linearized operator $d\tilde{F}[0]$ to conclude smoothness and finite-dimensionality of the moduli space $\mathcal{M}_L$.

Experimental results

Research questions

• RQ1 How can the deformation theory of special Lagrangian submanifolds be extended to include both conically singular and asymptotically conical ends?
• RQ2 What is the dimension of the moduli space of special Lagrangian conifolds in $\mathbb{C}^m$, and how does it depend on the decay rates at AC ends and the singularity type at CS points?
• RQ3 How do topological and analytic data—such as cohomology and restriction maps—contribute to the dimension of the moduli space?
• RQ4 Can the moduli space of SL conifolds be shown to be smooth and finite-dimensional under appropriate weighted Sobolev estimates?
• RQ5 How does the dimension of the moduli space change when the weight $\lambda$ crosses critical values, particularly when $\lambda > 0$?

Key findings

• The moduli space $\mathcal{M}_L$ of special Lagrangian conifolds is a smooth, finite-dimensional manifold when the linearized deformation operator is an isomorphism.
• The dimension of $\mathcal{M}_L$ is given by $\dim(\widetilde{H}_{0,\bullet}) + i(\Delta_{\boldsymbol{\mu},\boldsymbol{\lambda}}) - i(\Delta_{\boldsymbol{\mu},\boldsymbol{\lambda}'})$, where $i$ denotes the Fredholm index of the weighted Laplacian.
• For a stable SL cone $\mathcal{C} \subset \mathbb{C}^m$ with connected link, $\mathcal{M}_{\mathcal{C}}$ has dimension 0 when $\lambda \in (2-m, 0)$, indicating rigidity under such deformations.
• When $\lambda \in (1,2)$, $\mathcal{M}_{\mathcal{C}}$ has dimension $2m$, corresponding to the $2m$-dimensional family of translations of $\mathcal{C}$ in $\mathbb{C}^m$.
• For $\lambda \in (0,1)$, $\mathcal{M}_{\mathcal{C}}$ has dimension 0, showing that no non-trivial deformations exist in this range despite analytic freedom.
• The contribution of AC ends is captured via compactly supported cohomology, while CS singularities contribute via the vanishing of a restriction map, as formalized in Corollary 2.18 and Remark 2.19.