On the structure of complete $G_2$-solitons

Abstract: In this work, we establish compactness theorems for complete gradient $G_2$-solitons under the assumptions of a lower bound on the scalar curvature and a broad growth condition on the potential function associated with the gradient vector field. After first proving Gromov-Hausdorff convergence for such sequences, we sharpen this result by deriving epsilon-regularity estimates. As a consequence, we obtain smooth convergence provided there is a uniform energy bound at half the dimension.

• The paper introduces a systematic analytic framework and moduli classes for complete gradient G₂-solitons, establishing conditions for smooth and singular convergence.
• It proves pointed measured Gromov-Hausdorff compactness and C¹,α regularity, extending techniques from Ricci soliton theory to the G₂ context.
• Utilizing epsilon-regularity and L^(7/2) curvature bounds, the work shows that no curvature singularities develop in dimension seven, yielding conifold limits.

Compactness and Structure Theory for Complete $G_2$-Solitons

Introduction and Background

The analysis of geometric flows on closed $G_2$-structures in dimension seven, especially the $G_2$-Laplacian flow, has led to growing interest in self-similar solutions termed $G_2$-solitons. These play the role of singularity models, analogous to Ricci solitons in Ricci flow. A $G_2$-soliton is defined by a 7-manifold $(M,\varphi,g,X)$, where $\varphi$ is a closed $G_2$-structure, $g$ the associated metric, $X$ a vector field, and a constant $G_2$0 satisfying the equation

$G_2$1

Gradient $G_2$2-solitons are characterized by $G_2$3 for some smooth function $G_2$4. These special structures generalize the notion of torsion-free $G_2$5-manifolds and are intimately related to the analysis of geometric flows, moduli problems, and singularity formation.

The paper provides a systematic structure and compactness theory for complete gradient $G_2$6-solitons, specifically establishing conditions under which moduli spaces can be compactified, regularity of limits is controlled, and convergence to singular models is well-understood. These results are situated in the broader context of geometric analysis, drawing on analogy with Einstein metrics, Ricci solitons, and critical metric compactness.

Main Results and Technical Developments

Analytic Framework and Moduli Classes

A detailed analytic framework is developed for complete gradient $G_2$7-solitons. The classes $G_2$8 and its non-collapsed refinement $G_2$9 are introduced, parameterized by:

• Uniform lower bounds on scalar curvature.
• Controlled growth of the potential function $G_2$0: $G_2$1 for a smooth baseline function $G_2$2.
• Non-collapsing lower bounds via a local Perelman $G_2$3-entropy bound.

These conditions subsume explicit noncompact examples, including those constructed by Fowdar, Haskins-Nordström, Fino-Raffero, and others, with varying volume growth and potential behaviors.

Compactness Theorems

Pointed Measured Gromov-Hausdorff Compactness:

Sequences of complete gradient $G_2$4-solitons satisfying these analytic conditions admit subsequences converging in the pointed measured Gromov-Hausdorff topology to a limit metric measure space which decomposes into regular and singular parts. The regular part is characterized by convexity and tangent cone continuity, following the general theory of lower Ricci (or Bakry-Émery Ricci) bounds [4234100].

$G_2$5 Regularity and Stronger Compactness:

Imposing an additional non-collapsing entropy bound, the convergence can be promoted to pointed $G_2$6, yielding a limit singular space $G_2$7 smooth away from a singular set of codimension at least four. This extends Cheeger-Naber/Epsilon-regularity/Anderson-type theory to the $G_2$8-soliton context.

Smooth and Conifold Limits:

An $G_2$9-regularity theorem is proved: if local entropy is sufficiently small (depending only on dimension), all singularities are absent and convergence is $G_2$0. Similarly, if the sequence exhibits a uniform local $G_2$1-bound on the Riemannian curvature, limits are smooth and hence are conifold $G_2$2-solitons in the sense of Chen-Wang.

Strong Claims:

A central claim is that no curvature singularities appear in the $G_2$3-compactness under an $G_2$4-energy bound, in sharp contrast to, e.g., the case of Einstein metrics in dimension $G_2$5 where orbifold singularities may appear.

Regularity and Analytical Structure

Through an intricate system of tensor identities, Weitzenböck formulas, and decomposition theorems for the torsion, the authors develop fine analytic tools. They systematically relate all higher derivatives of the $G_2$6-structure and its torsion to those of the Riemann curvature, proving domination and equivalence estimates at all orders (using multi-index notation), similar to the structure for Ricci and Einstein solitons. The bootstrapping argument, combining $G_2$7 estimates, the soliton equation, and Schauder-type regularity, shows that on the regular part, all $G_2$8-structures, metrics, and potentials are $G_2$9.

The derivation and application of Laplacian comparison theorems, volume doubling, local Sobolev inequalities, and implementation of Perelman-type entropy rigidity are essential in the analysis. The use of metric measure techniques and the Bakry-Émery Ricci tensor connects the argument to general convergence and stability theory.

Examples and Sharpness

Multiple explicit complete (noncompact) gradient $G_2$0-soliton solutions are analyzed—cohomogeneity-one, homogeneous, shrinking, steady, and expanding—invariant under substantial symmetry groups. These exhibit diverse behaviors for curvature and potential function growth, serving as both models for the compactness theory and as a warning about the range of possible limits.

A strong consequence is the absence of compact, non-torsion-free (non-trivial) $G_2$1-solitons, and a full classification in the compact homogeneous case. This is established by integrating the soliton equations and exploiting curvature positivity, extending Lin's and Lotay-Wei's nonexistence results.

Implications and Directions

Theoretical Impact

This work fills a gap in the understanding of singularity formation (moduli compactness) for geometric flows on $G_2$2-manifolds, providing a systematic structure theory for $G_2$3-solitons. The development of epsilon-regularity and gap theorems in this context has direct analogies to Einstein, Bach-flat, and Ricci soliton moduli theories and sets out a template for further exploration in special holonomy and calibrated geometry.

A notable claim is that under the $G_2$4 Riemann curvature bound, all singularities disappear in dimension 7; this is sharper than the orbifold compactness known in lower-dimension Einstein geometry.

The results also identify precise analytic obstructions and sharp growth rates for the potential, relating the regularity and collapse properties directly to entropy and energy. The use of Perelman's entropy in this context is an important generalization of Ricci flow techniques.

Practical and Further Research

The results have consequences for the construction of moduli spaces of $G_2$5-solitons, the analysis of singularity models in $G_2$6-Laplacian flows, understanding formation of singular sets (or their absence), and the analytic underpinnings of calibrated geometry. The techniques extend naturally to other geometric flows, including those on Spin(7), Calabi–Yau, and various critical metrics.

This framework can guide future investigations on:

• The explicit description of all possible singular models in the absence of global energy bounds.
• The extension of these compactness results to flows with less regular or less symmetric structures.
• Connections between entropy functional rigidity and geometric compactness more broadly, including singularity formation in Ricci and mean curvature flow.
• Possible generalizations to more general geometric structures and flows with extra symmetries or constraints.

Conclusion

The paper establishes a robust, highly-structured analytic theory for complete gradient $G_2$7-solitons, characterized by pointed measured Gromov-Hausdorff compactness, strong regularity of limits, and new epsilon-regularity theorems. The results parallel the best in the field of critical metrics and geometric flows while pointing towards further developments in geometric analysis, singularity theory, and calibrated geometry. The approach provides a template for moduli compactness theory for non-Ricci flows with constrained holonomy and complex algebraic/topological structure.

Reference:

"On the structure of complete $G_2$8-solitons" (2606.06619)