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A Liouville Theorem and $C^α$-Estimate for Calabi-Yau Cones

Johan Jacoby Klemmensen

TL;DR

The paper proves a Liouville theorem for cscK metrics on Calabi–Yau cones, showing any metric uniformly equivalent to the cone metric is obtained from the cone metric by a cone automorphism commuting with scaling. It then establishes a Krylov-inspired $C^{0,lpha}$-type Schauder estimate near the apex for Kähler metrics with bounded scalar curvature, yielding polynomial convergence to the cone metric with rate $r^{lpha}$ up to a holomorphic automorphism. The approach combines rescaling and tangent-cone analysis, heat-kernel bounds, and Evans– Krylov-type regularity for complex Monge–Ampère equations, to conclude both Liouville-type rigidity and apex regularity without requiring smoothability. As corollaries, the tangent cones are unique and the metrics are asymptotic to the Calabi–Yau cone with explicit decay, under uniform equivalence. These results extend Liouville-type rigidity to singular Calabi–Yau cones and provide robust apex-regime regularity in a conical setting.

Abstract

Let $(\mathscr{C}, ω_{\mathscr{C}})$ be a Ricci-flat, simply connected, conical Kähler manifold. We establish a Liouville theorem for constant scalar curvature Kähler (cscK) metrics on $\mathscr{C}$. The theorem asserts that any cscK metric $ω$ satisfying the uniform bound $\frac{1}{C} ω_{\mathscr{C}} \leq ω\leq C ω_{\mathscr{C}}$ for some $C\geq1$ is equal to $ω_{\mathscr{C}}$ up to a holomorphic automorphism that commutes with the scaling action of the cone structure. Next, we develop a $C^{0,α}$-estimate for uniformly bounded Kähler metrics on a ball around the apex, using a Hölder-type seminorm inspired by Krylov. This estimate applies for small $α> 0$ under the assumption of uniformly bounded scalar curvature. As a corollary of this result, we show that such a Kähler metric $ω$ is asymptotic to the Ricci-flat cone metric $ω_{\mathscr{C}}$, with polynomial decay rate $r^α$ and for sufficiently small $α> 0$.

A Liouville Theorem and $C^α$-Estimate for Calabi-Yau Cones

TL;DR

The paper proves a Liouville theorem for cscK metrics on Calabi–Yau cones, showing any metric uniformly equivalent to the cone metric is obtained from the cone metric by a cone automorphism commuting with scaling. It then establishes a Krylov-inspired -type Schauder estimate near the apex for Kähler metrics with bounded scalar curvature, yielding polynomial convergence to the cone metric with rate up to a holomorphic automorphism. The approach combines rescaling and tangent-cone analysis, heat-kernel bounds, and Evans– Krylov-type regularity for complex Monge–Ampère equations, to conclude both Liouville-type rigidity and apex regularity without requiring smoothability. As corollaries, the tangent cones are unique and the metrics are asymptotic to the Calabi–Yau cone with explicit decay, under uniform equivalence. These results extend Liouville-type rigidity to singular Calabi–Yau cones and provide robust apex-regime regularity in a conical setting.

Abstract

Let be a Ricci-flat, simply connected, conical Kähler manifold. We establish a Liouville theorem for constant scalar curvature Kähler (cscK) metrics on . The theorem asserts that any cscK metric satisfying the uniform bound for some is equal to up to a holomorphic automorphism that commutes with the scaling action of the cone structure. Next, we develop a -estimate for uniformly bounded Kähler metrics on a ball around the apex, using a Hölder-type seminorm inspired by Krylov. This estimate applies for small under the assumption of uniformly bounded scalar curvature. As a corollary of this result, we show that such a Kähler metric is asymptotic to the Ricci-flat cone metric , with polynomial decay rate and for sufficiently small .

Paper Structure

This paper contains 37 sections, 51 theorems, 307 equations.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Structure of the Paper
  4. A Liouville Theorem for Calabi-Yau Cones
  5. From Scalar Curvature to the Complex Monge-Ampère Equation
  6. A Liouville Theorem for the Tangent Cones at $\infty$ and $o$
  7. Cone Structure of the Asymptotic Limits
  8. The Tangent Cones and $\omega_{\mathscr{C}}$ have equal Reeb fields
  9. Commuting Reeb Fields
  10. Reeb fields are Killing fields
  11. Deformations of the Sasakian Structure
  12. The Tangent Cones equal $\omega_{\mathscr{C}}$
  13. Combining Tangent Cones to Liouville Theorem
  14. $C^{0,\alpha}$-Type Estimate for Kähler Metrics in a Neighborhood of $o$
  15. $C^{0,\alpha}$-Type Seminorm
  16. ...and 22 more sections

Key Result

Theorem 1

Let $(\mathscr{C},\omega_\mathscr{C})$ be a Calabi-Yau cone. Take any cscK metric $\omega$ on $\mathscr{C}$ such that there exists some constant $C>0$ with Then $\omega = \Psi^*\omega_\mathscr{C}$ for some cone automorphism $\Psi\in \mathop{\mathrm{Aut}}\nolimits_\mathop{\mathrm{Scl}}\nolimits(\mathscr{C})$.

Theorems & Definitions (102)

  • Theorem 1: Theorem \ref{['LiouvilleTheorem']}
  • Theorem 2: Theorem \ref{['TheoremHolderBound']}
  • Corollary 3: Corollary \ref{['CorollaryOfCalpha']}
  • Definition 2.1
  • Theorem 2.3: Liouville Theorem for Calabi-Yau cones
  • Definition 2.5: heinHigherorder2020
  • Lemma 2.6
  • proof
  • Remark 2.7
  • Proposition 2.8
  • ...and 92 more