Special Lagrangian Three-Tori

• Special Lagrangian three-tori are three-dimensional tori embedded in Calabi–Yau threefolds that serve as calibrated Lagrangian submanifolds with minimal volume and local rigidity.
• Gluing constructions, notably via Lawlor necks, enable desingularization of intersecting tori and ensure uniqueness up to translation symmetry under analytic controls.
• Analytic techniques using weighted Hölder norms and displacement energy provide rigorous tools for understanding moduli spaces and mirror symmetry phenomena in calibrated geometry.

A special Lagrangian three-torus is a three-dimensional torus embedded as a calibrated Lagrangian submanifold in a Calabi–Yau threefold, distinguished by the property that the real part of the ambient holomorphic volume form restricts to its volume form. Such submanifolds represent a central object in calibrated geometry, mirror symmetry, and geometric analysis due to their minimality, local rigidity, and moduli-theoretic importance. The construction and classification of special Lagrangian three-tori involve an overview of techniques from differential geometry, analysis of PDEs, and symplectic topology.

1. Analytical and Geometric Characterization

Let $(X, \omega, \Omega)$ denote a Calabi–Yau threefold, where $\omega$ is a Kähler form and $\Omega$ is a holomorphic $(3,0)$-form. An embedded torus $L \subset X$ is special Lagrangian (SLag) if it satisfies: $$\omega|_L = 0, \quad \mathrm{Im}(e^{-i\theta} \Omega)|_L = 0$$ for some phase $\theta$. Equivalently,

$$\mathrm{Re}(e^{-i\theta} \Omega)|_L = \mathrm{vol}_L$$

so $L$ is calibrated by $\mathrm{Re}(e^{-i\theta} \Omega)$.

In flat models such as $(T^6, \omega_0, J_0, \Omega_0)$ with $\Omega_0 = dz_1 \wedge dz_2 \wedge dz_3$, flat special Lagrangian three-tori minimize volume in their homology class and are stable under deformations by McLean's theory. Their moduli space is locally modeled by $H^1(L, \mathbb{R})$.

2. Gluing and Desingularization via Lawlor Neck

The gluing construction produces new special Lagrangian submanifolds by resolving transverse intersections of existing SLags. Given two flat special Lagrangian tori $M_1, M_2 \subset T^6$ meeting transversally at $P$, a neighborhood of $P$ in $M_1 \cup M_2$ is excised, and a model Lawlor neck is inserted. The neck is constructed as an explicit normal graph over an annular domain

$$A(b_0, b_1; X) = \{a \in \mathbb{R}^6 \setminus \{0\} : b_0 < |a| < b_1,\; a/|a| \in X\}$$

where $X \subset S^5$ is a minimal link, and the submanifold takes the form

$M = G(v) = \{a + v(a) : a \in A(b_0, b_1; X)\}$

with $v$ a normal vector field. The process controls $v$ via weighted Hölder norms and uses compactness and asymptotic lemmas (e.g., Allard, Simon) to ensure the glued SLag is smooth and remains close to the Lawlor neck in a tubular neighborhood of $P$.

3. Uniqueness Theorem for Gluing Special Lagrangian Tori

In the specific case of gluing flat special Lagrangian tori in $(T^6, \omega_0, J_0, \Omega_0)$, a uniqueness theorem establishes that—under suitable smallness and asymptotic conditions on the graph representation—any special Lagrangian submanifold constructed via this gluing is unique modulo the translation symmetry of the torus. Precisely, if a candidate SLag is $C^1$-close to the Lawlor neck near $P$ and matches the flat tori outside the gluing region, then it must be the model glued SLag up to a translation. The proof deploys analytic uniqueness in annular regions using Simon’s asymptotic method for minimal submanifolds, patched with normal vector field estimates.

4. Generalizations, Tropical and Toric Constructions

Special Lagrangian three-tori arise in broader settings such as toric Sasaki manifolds, Calabi–Yau cones, and symplectic toric varieties. In toric Kähler cones, SLags are constructed as fixed-point loci of anti-holomorphic involutions and then moved by Hamiltonian torus actions and dilations: $F(p, t) = \rho(t)\cdot T(t)\cdot p$ with $T(t) = \exp(f(t)\xi)$ and $p$ in the canonical real form $C(S)^\sigma$. In cones of real dimension six, explicit examples of SLag submanifolds diffeomorphic to $E_g \times \mathbb{R}$ (with $E_g$ a surface of genus $g$) and compact self-shrinkers $E_g \times S^1$ are constructed (Yamamoto, 2012).

Via tropical-to-Lagrangian correspondence, rational tropical curves inside moment polyhedra of toric varieties are "lifted" to Lagrangian 3-manifolds whose topology (e.g., rational homology sphere, graph-manifold decompositions) is governed by tropical multiplicities, establishing an enumerative link between tropical and Lagrangian geometry (Mikhalkin, 2018).

5. Moduli, Rigidity, and Exotic Phenomena

Special Lagrangian tori are subject to subtle moduli phenomena in symplectic geometry. Families of exotic (non-monotone) Lagrangian tori exist in high-dimensional symplectic manifolds, distinguished up to symplectomorphism but Lagrangian isotopic with identical classical invariants (area and Maslov class) (Brendel, 2023). The key invariant that discriminates these is the displacement energy germ, computable in Darboux charts and robust under bubbling phenomena. In explicit constructions, such as

$$T(a_1, a_2, a_3) = S^1(a_1) \times S^1(a_2) \times S^1(a_3)$$

the displacement energy is $e(T(a_1, a_2, a_3)) = \min\{a_1, a_2, a_3\}$, but for exotic tori, combinatorial details of reductions encode additional invariants.

In the context of SYZ mirror symmetry and Calabi–Yau transitions, special Lagrangian tori and spheres are exchanged via topological transitions; for instance, conifold transitions trade holomorphic 2-cycles for special Lagrangian 3-cycles, altering Betti numbers and the moduli-theoretic structure (Collins et al., 2021).

6. Special Lagrangian Tori in Fibrations and Collapsing Calabi–Yau Manifolds

Special Lagrangian tori play a critical role in torus fibrations of Calabi–Yau threefolds, especially in the Strominger–Yau–Zaslow (SYZ) picture. In noncompact Tian–Yau settings or collapsing K3-fibered Calabi–Yau threefolds, special Lagrangian tori arise as fibers of singular fibrations, often constructed via Lagrangian mean curvature flow starting from model Lagrangians with controlled geometry (Collins et al., 2019, Chiu et al., 23 Oct 2024). The analysis involves gluing local model solutions in fibers (e.g., using weighted Hölder spaces) and deforming via contraction mapping arguments to exact calibrated solutions.

Admissible loops in the base of a K3-fibration allow for mapping-torus constructions, which, when the fiber admits a special Lagrangian two-torus, yield higher-dimensional SLags diffeomorphic to $T^3$. The techniques developed for vanishing cycles and spheres in these settings are directly applicable to three-tori via adjusting the topological input.

7. Connections to Mirror Symmetry and Mathematical Physics

Special Lagrangian three-tori serve as probes for the structure of moduli spaces and wall-crossing phenomena in mirror symmetry. Their abundance, rigidity, and degenerations (e.g., convergence to singular immersions) encode deep aspects of the SYZ conjecture and its categorical formulations. In transition phenomena, their role as calibrated representatives reflects the exchange of physical states under dualities, linking the geometry of the ambient Calabi–Yau manifold to gauge-theoretic and enumerative invariants.

The explicit uniqueness and gluing results for flat SLag three-tori (Imagi, 2011) provide a rigorous foundation for the desingularization of torus fibrations, the resolution of singular fibers, and the analytic control necessary for compositional SYZ-type constructions.

Summary Table: Principal Constructions and Phenomena

Construction Type	Topology	Key Techniques
Lawlor neck gluing	T³ via connection of tori	Annular graphs, weighted Hölder norms, uniqueness
Toric cone examples	E_g × ℝ, E_g × S¹	Torus actions, involutions, explicit moment maps
Exotic tori	T³ (isotopic, non-symplecto)	Symplectic reduction, displacement energy germ
Fibration fibers	T³ fiber in torus fibration	LMCF, gluing, contraction mapping
Tropical lifts	Graph-manifold, rational sphere	Tropical-to-Lagrangian correspondence

Special Lagrangian three-tori embody core aspects of calibrated geometry and mirror symmetry, providing technical exemplars for metric, analytic, and enumerative studies in higher-dimensional geometry. Their construction and uniqueness are governed by geometric analysis, topological reduction, and symplectic invariants, making them central objects in the modern geometric landscape.