Complete noncompact G2-manifolds with ALC asymptotics

Abstract: We prove existence, uniqueness and structure results for complete noncompact 7-dimensional G2-holonomy metrics with ALC (asymptotically locally conical) asymptotics. We regard such spaces as G2-analogues of ALF gravitational instantons in 4-dimensional hyperkähler geometry. Our main results include the existence of a G2-analogue of the Atiyah-Hitchin metric in 4-dimensional hyperkähler geometry, the existence of a good moduli theory for ALC G2-holonomy metrics and rigidity results for ALC G2-metrics in terms of the symmetries of their asymptotic model. The analytic toolkit needed to prove all these results is a robust Fredholm theory for the natural geometric linear elliptic operators on ALC spaces. We provide a self-contained derivation of this Fredholm theory for arbitrary Riemannian manifolds with ALC asymptotics. Since our ALC Fredholm theory does not rely on imposing any holonomy reduction or curvature conditions it may also be of utility beyond the setting of ALC special holonomy metrics. As one such application of our general Fredholm theory we prove some Hodge-theoretic results on general ALC spaces.

• The paper introduces dihedral-type ALC G₂ metrics on S³ × O(-4), confirming predictions from the physics literature.
• It develops a robust analytic framework based on Fredholm theory for Dirac-type operators on ALC spaces.
• The work establishes rigidity and unobstructed deformation results, which enhance understanding of special holonomy geometry.

Complete Noncompact G₂-Manifolds with ALC Asymptotics

Introduction and Motivation

This work establishes existence, uniqueness, analytic, and geometric structure results for complete noncompact 7-manifolds with holonomy $G_2$ and asymptotically locally conical (ALC) asymptotics (2604.14704). Paralleling the role of ALF (asymptotically locally flat) metrics in 4-dimensional hyperkähler geometry, these spaces extend the landscape of noncompact Ricci-flat and special holonomy metrics into higher dimensions, with ALC 7-manifolds identified as $G_2$-analogues of ALF gravitational instantons. Two chief lines of investigation are pursued: drawing analogies and extending results from ALF/ALG geometry in dimension four to the $G_2$ ALF/ALC case; and extending deformation and moduli theory from the compact and asymptotically cylindrical/conical $G_2$-manifolds to metrics with ALC ends.

Main Results

The principal advances are threefold:

1. Existence of Dihedral-Type ALC $G_2$ Metrics: The first construction of complete, noncompact ALC $G_2$-holonomy metrics of dihedral type is achieved, specifically a $1$-parameter family on $S^3 \times \mathcal{O}_{\mathbb{P}^1}(-4)$. This confirms a conjectured family ($\mathbb{A}_7$) from the physics literature, and provides the first such instances beyond the infinitely many known cyclic-type ALC examples.

2. Structure and Rigidity Theorems for ALC Ends: It is proved that every ALC $G_2$ end metric is asymptotic (with exponentially decaying error) to a locally circle-invariant $G_2$0 structure, and that, in the cyclic case, a global structure-preserving circle action exists. Further, a rigidity result analogous to the "positive mass theorem" in the ALF context is established: for cyclic-type ALC $G_2$1 manifolds with principal circle bundles over Calabi–Yau cones, any such manifold is finitely covered by $G_2$2 an AC Calabi–Yau 3-fold.

3. Fredholm Theory and Deformation Theory in the ALC Setting: A detailed, self-contained analytic framework for elliptic operators—specifically Dirac-type operators—on ALC spaces is constructed. This framework bypasses microlocal techniques and is based directly on the geometric structure of ALC ends. The analytic machinery underpins an unobstructed deformation theory for ALC $G_2$3 metrics, generalizing Joyce’s theory for the compact case to ALC noncompact settings. The associated moduli space of torsion-free ALC $G_2$4 metrics is shown to be a smooth manifold locally modeled on closed and coclosed 3-forms of appropriate decay.

The analytic toolkit, based on weighted Sobolev and Hölder spaces and a robust Fredholm theory, extends naturally to higher-rank torus fibrations and Seifert ALC ends, and provides new algebraic-topological computations (e.g., of weighted $G_2$5 cohomology and jumps at indicial roots) for general metric ends of this type.

Analytic Framework and Hodge Theory

The direct analytic approach, focusing on model operators at infinity and their perturbations, yields a Fredholm theory for admissible Dirac-type operators, with index jump formulas precisely tracking variations as decay rates cross indicial roots of the model cones. The theory incorporates:

• Fourier decomposition along fibres, leveraging the collapse of the asymptotic geometry to control higher modes.
• Adapted connections (following Bismut), which better reflect the asymptotic bundle structure than the Levi-Civita connection.
• Generalization to fibred boundary metrics with higher-rank torus fibres and Seifert fibrations of orbifold type.

In an application to Hodge theory, all contributions to spaces of closed and coclosed $G_2$6-forms of a given decay rate are computed in terms of topological data, with technically refined behavior at non-$G_2$7-integrable rates and in the presence of conical singularities or exceptional cross-sections. This recovers and extends results of Hausel–Hunsicker–Mazzeo on $G_2$8-cohomology without recourse to microlocal analysis.

Deformation and Moduli Space Structure

The moduli space of torsion-free ALC $G_2$9-structures with fixed (sufficiently fast) decay rate to a base ALC end is unobstructed and smooth, with tangent space isomorphic to closed and coclosed 3-forms of the same decay. Cohomological and analytic computations give precise moduli space dimension counts in explicit examples, closely paralleling computations for the AC and ACyl cases. For metrics constructed via circle bundles over AC Calabi–Yau 3-folds, the moduli space is shown to be locally diffeomorphic to the moduli space of AC Calabi–Yau structures (satisfying a topological constraint) on the base.

Rigidity and Symmetry Inheritance: Under a natural injectivity assumption on the natural map from decaying harmonic 3-forms to de Rham cohomology, all automorphisms at infinity (including discrete symmetries and group actions) extend to the entire manifold, leading to uniqueness results for highly symmetric ALC $G_2$0 metrics.

Desingularization and New Examples

A gluing and desingularization theory for ALC $G_2$1 manifolds with isolated conical singularities is developed, adapting techniques from compact conically singular $G_2$2-spaces (Karigiannis) and dihedral ALF gravitational instantons (Biquard–Minerbe). This machinery is applied to realize the predicted dihedral-type metrics and to analyze large circle-length limits, in which collapsed ALC geometries degenerate to orbifold or conically singular models, with subsequent smoothing via analytic gluing.

Analytic subtleties arise in the solution of Poisson equations on weighted spaces, particularly for critical rates associated with double indicial roots (notably, for Laplacians on 2-forms on 6-dimensional cones).

Implications and Future Directions

This work demonstrates that the moduli and analytic theories for ALC $G_2$3 metrics are as robust as their ACyl and AC counterparts, and that the ALC setting admits rich new phenomena, such as the dichotomy between cyclic and dihedral asymptotics paralleling the ALF case, and finer rigidity constraints related to symmetry extension and the topology of the ends. Practically, the construction of dihedral-type metrics fills in an important class of "building blocks" for future constructions of compact $G_2$4-manifolds via collapse and gluing (analogous to the role played by the Atiyah–Hitchin and Taub–NUT spaces in dimension four), and broadens the dictionary connecting differential geometry, special holonomy, and string-theoretic / M-theory compactifications.

The analytic Fredholm theory provided here, being independent of holonomy or curvature assumptions, may be applied beyond the context of $G_2$5 holonomy; in particular, to further understanding of collapse, moduli, and boundary value problems for more general complete noncompact Ricci-flat or special holonomy spaces. Prospective research directions include refinements in the construction and classification of ALC (and more general fibred boundary) special holonomy manifolds, further desingularization and collapse scenarios, and applications to gauge theory and mathematical physics (in particular, compactifications yielding lower-dimensional effective field theories with rich moduli).

Conclusion

This work fully develops both the analytic and geometric structure theory for complete noncompact $G_2$6-holonomy metrics with ALC asymptotics, establishing the first dihedral-type metrics, a robust Fredholm theory for model elliptic operators, and detailed moduli space computations. It both deepens the analogy between 4-dimensional ALF and 7-dimensional ALC geometries and provides essential new tools and families for future study of collapse, desingularization, and the global geometry of exceptional holonomy manifolds (2604.14704).