[Paper Review] Deformations of Lagrangian Type Submanifolds inside G2 manifolds

Rebecca J. Glover, Sema Salur|arXiv (Cornell University)|Mar 10, 2015

Geometric Analysis and Curvature Flows10 references3 citations

TL;DR

This paper studies deformations of Lagrangian-type submanifolds in G₂ manifolds, introducing a new class of 4-dimensional RS submanifolds. It proves that the space of infinitesimal deformations of a compact, orientable HL 3-fold is isomorphic to the direct sum of smooth functions and closed 2-forms on the submanifold, while for RS 4-folds, it is isomorphic to closed 3-forms, generalizing deformation theory in special holonomy geometry.

ABSTRACT

3-dimensional Harvey Lawson submanifolds were introduced in an earlier paper by Akbulut-Salur, as examples of Lagrangian-type manifolds inside G2 manifold. In this paper, we first show that the space of deformations of a smooth, compact, orientable Harvey-Lawson submanifold HL in a G2 manifold M can be identified with the direct sum of the space of smooth functions and closed 2-forms on HL. We then introduce a new class of Lagrangian-type 4-dimensional submanifolds inside G2, call them RS submanifolds and prove that the space of deformations of a smooth, compact, orientable RS submanifold in a G2 manifold M can be identified with closed 3-forms on RS.

Motivation & Objective

• To extend the theory of Lagrangian-type submanifolds in G₂ manifolds beyond associative and coassociative submanifolds.
• To define and study a new class of 4-dimensional submanifolds, called RS submanifolds, analogous to Lagrangian submanifolds in symplectic geometry.
• To characterize the space of infinitesimal deformations of these submanifolds within their respective classes.
• To establish a deformation theory for submanifolds calibrated by the G₂ 3-form φ and its Hodge dual ⋆φ.
• To explore connections between G₂ geometry, calibrated submanifolds, and mirror symmetry via Fukaya categories.

Proposed method

• Define Harvey-Lawson (HL) submanifolds as 3-dimensional submanifolds where the G₂ 3-form φ vanishes when restricted to them.
• Introduce RS submanifolds as a new class of 4-dimensional Lagrangian-type submanifolds calibrated by the Hodge dual 4-form ⋆φ.
• Use the exponential map and linearization of the deformation map F to analyze infinitesimal deformations of submanifolds.
• Compute the linearization dF(0) of the deformation map F: Γ(N(RS)) → Λ⁴T*(RS), showing it equals d⋆v = ⋆(d*v) for the dual 1-form v of the normal vector field V.
• Identify the kernel of dF(0) with closed 3-forms on the RS submanifold, establishing the deformation space.
• Leverage the cross product and triple cross product structures on G₂ manifolds to define the geometric framework for HL and RS submanifolds.

Experimental results

Research questions

• RQ1 What is the space of infinitesimal deformations of a compact, orientable 3-dimensional Harvey-Lawson submanifold in a G₂ manifold?
• RQ2 How can a new class of 4-dimensional Lagrangian-type submanifolds (RS submanifolds) be defined in G₂ manifolds?
• RQ3 What is the deformation space of RS submanifolds, and how does it relate to differential forms on the submanifold?
• RQ4 Can the deformation theory of HL and RS submanifolds be generalized beyond torsion-free G₂ structures?
• RQ5 How do the geometric structures of cross products and frame fields on G₂ manifolds facilitate the definition and analysis of these submanifolds?

Key findings

• The space of infinitesimal deformations of a smooth, compact, orientable 3-dimensional Harvey-Lawson submanifold in a G₂ manifold is isomorphic to the direct sum of the space of smooth functions and the space of closed 2-forms on the submanifold.
• The space of infinitesimal deformations of a smooth, compact, orientable 4-dimensional RS submanifold in a G₂ manifold is isomorphic to the space of closed 3-forms on the submanifold.
• The deformation space for HL submanifolds is infinite-dimensional, reflecting the non-compactness of the moduli space.
• The deformation space for RS submanifolds is also infinite-dimensional, parameterized by closed 3-forms.
• The results hold not only for G₂ manifolds with torsion-free structure but also for manifolds with closed or co-closed G₂ structures.
• The geometric role of the cross product in HL submanifolds and the triple cross product in RS submanifolds mirrors the role of an almost complex structure in symplectic Lagrangian submanifolds.