A Liouville theorem for some asymptotically conical Calabi-Yau manifolds

Abstract: Let $(\mathcal{C}, J_{\mathcal{C}}, ω{\mathcal{C}}, g{\mathcal{C}})$ be a Calabi-Yau cone and $(M, J, ω, g)$ an open Ricci-flat Kähler manifold. We show that, if there exists a diffeomorphism $Φ: \mathcal{C} \setminus \overline{B_1(o)} \rightarrow M \setminus K$, for some compact $K \subset M$, such that $Φ^{*}J$ is asymptotic to $J_{\mathcal{C}}$ and $C^{-1} ω{\mathcal{C}} \leq Φ^{*} ω\leq C ω{\mathcal{C}}$ for some $C \geq 1$, then $(M, g)$ is asymptotically conical (AC) with tangent cone at infinity given by $(\mathcal{C}, d_{g_{\mathcal{C}}})$. As a consequence, we obtain that any Ricci-flat Kähler metric on $T^{*}S^n$ which is quasi-isometric to the Stenzel metric must be equal to the Stenzel metric up to scaling and diffeomorphism. Similarly, any Ricci-flat Kähler metric on $\mathcal{O}_{\mathbb{P}^1}(-1)^{\oplus2}$ which is quasi-isometric to the Candelas-De la Ossa metric must be equal to the Candelas-De la Ossa metric up to scaling and diffeomorphism. This provides new examples of complete Calabi-Yau manifolds for which a Liouville-type theroem holds.

• The paper establishes a Liouville-type theorem proving that Ricci-flat Kähler metrics on AC Calabi–Yau manifolds are uniquely determined, up to scaling and diffeomorphism.
• Using blow-down rescalings and C^(k,β) convergence, the analysis demonstrates quadratic curvature decay and Gromov–Hausdorff convergence to the Calabi–Yau cone.
• The approach extends classical Liouville results to noncompact settings, influencing moduli classification and applications in singular Kähler–Einstein geometry.

Liouville-Type Rigidity for Asymptotically Conical Calabi–Yau Manifolds

Introduction and Motivation

The paper "A Liouville theorem for some asymptotically conical Calabi-Yau manifolds" (2606.04213) addresses the rigidity of Ricci-flat Kähler metrics on open manifolds that are asymptotic, in a strong geometric sense, to Calabi–Yau cones. Rigidity theorems of Liouville-type have played a key role in complex differential geometry, especially regarding regularity and uniqueness for solutions of complex Monge–Ampère equations. The classical Liouville result for $\mathbb{C}^n$ provides that a Kähler metric with constant determinant and global quasi-isometry to the flat metric must be flat up to an automorphism. This result has been extended in various directions, including for product and conical geometries.

The purpose of this paper is to establish an analogue of the Liouville theorem for open Ricci-flat Kähler manifolds that are sufficiently close, in both metric and complex structure, to Calabi–Yau cones outside compact subsets: namely, that such manifolds must be globally asymptotically conical, and further, that any quasi-isometric Ricci-flat Kähler metric on certain AC Calabi–Yau manifolds is unique up to scaling and diffeomorphism.

Preliminaries and Definitions

The analysis is situated in the context of Riemannian cones $(C, g_C)$ with a parallel complex structure $J_C$, for which the metric is Kähler and Ricci-flat—the so-called Calabi–Yau cones. Asymptotically conical (AC) manifolds are open Riemannian manifolds $(M, g)$ equipped with a diffeomorphism from the complement of a compact subset of $M$ to the exterior of a compact ball in $C$, such that the pulled-back metric converges to the cone metric at a polynomial rate. When $M$ carries a compatible Ricci-flat Kähler structure, it is called an AC Calabi–Yau manifold.

The survey in the paper details the latest progress in classification (notably by Conlon–Hein), and the critical Liouville theorem for Calabi–Yau cones established by Klemmensen, which asserts metric rigidity under quasi-isometry for complete Calabi–Yau cones. It also introduces the relevant moduli, automorphisms, and complex structures underpinning the rigidity statements.

The Main Theorem and Proof Structure

The principal result asserts that if $(C, J_C, \omega_C, g_C)$ is a Calabi–Yau cone, and $(M, J, \omega, g)$ is a complete Ricci-flat Kähler manifold such that, outside a compact set, the complex structure and Kähler form pulled back to the cone are close (in a precise $C^{k,\beta}$ topology and in a uniform quasi-isometry sense), then $(C, g_C)$0 is AC with tangent cone at infinity canonically $(C, g_C)$1.

This theorem is implemented via the following strategy:

1. Blow-down Rescalings and Convergence: Consider rescalings of the metric at infinity and extract $(C, g_C)$2 subsequential limits.
2. Liouville Rigidity for the Limit: Demonstrate, using Klemmensen’s Liouville theorem, that any such limiting metric must be pulled back from the underlying Calabi–Yau cone by an automorphism.
3. Quadratic Curvature Decay: Using analysis akin to Cheeger–Colding and Sun–Zhang, show that the metric must have quadratic curvature decay at infinity. This is essential for the asymptotically conical property and for applying classification results.
4. Gromov–Hausdorff Convergence: Establish pointed Gromov–Hausdorff convergence of the rescaled manifolds to the limiting cone, confirming uniqueness of the tangent cone at infinity.

As an explicit consequence, on both the Stenzel metric on $(C, g_C)$3 and the Candelas–de la Ossa metric on the small resolution $(C, g_C)$4 of a three-dimensional nodal cone, any Ricci-flat Kähler metric quasi-isometric to the canonical AC metric must agree with it up to a diffeomorphism and scaling. This provides strong rigidity for these families of AC Calabi–Yau manifolds.

Implications, Applications, and Prospects

The main theorem positions itself as a natural extension of classical Liouville results to highly nontrivial, noncompact and singular geometric settings. The notable implications are:

• Uniqueness: For a large class of AC Calabi–Yau manifolds—especially those arising as smoothings or crepant resolutions of nodal cones—there is effective uniqueness of the Ricci-flat Kähler metric under natural geometric constraints.
• Global Geometric Structure: The results confirm that large-scale geometry and complex structure behavior at infinity rigidly determine the global metric up to the intrinsic symmetries of the cone.
• Classification: Combined with the recent complete classification of AC Calabi–Yau manifolds by Conlon–Hein and the relevant uniqueness theorems, these Liouville-type results sharply restrict the moduli of possible Ricci-flat Kähler metrics in each asymptotic class.
• Regularity and Analysis of Monge–Ampère: The proof techniques emphasize the robustness of regularity and convergence results for complex Monge–Ampère equations under quasi-isometric control, with potential further applications to singular Kähler–Einstein geometry.
• Extensions: The analytic framework may generalize to other complex cones, allow less regular asymptotics, or inspire global uniqueness results for other classes of noncompact Ricci-flat manifolds.

Future research could focus on extending such rigidity to more general types of singularities, relaxing decay or asymptotic regularity hypotheses, or exploring applications to moduli problems in geometric analysis and mathematical physics, particularly in String Theory settings where AC Calabi–Yau manifolds frequently arise.

Conclusion

This work establishes a Liouville-type rigidity theorem for asymptotically conical Calabi–Yau manifolds, proving that Ricci-flat Kähler metrics with suitable geometric control at infinity must be globally asymptotically conical and are unique up to scaling and diffeomorphism on key examples. The approach, relying on a blend of analytic (complex Monge–Ampère theory, curvature decay) and geometric (asymptotic structure, classification) ingredients, affirms the determinative power of asymptotic data for the global geometry of noncompact Calabi–Yau spaces. These results reinforce foundational aspects of metric rigidity and offer a path to related statements in broader settings of noncompact Kähler and Einstein geometry.