A Liouville Theorem and $C^{\alpha}$-Estimate for Calabi-Yau Cones

TL;DR

This paper proves a Liouville theorem for constant scalar curvature K"ahler metrics on Calabi-Yau cones and establishes a $C^{ ext{alpha}}$-estimate showing asymptotic behavior of such metrics near the cone's apex.

Contribution

It introduces a Liouville theorem characterizing cscK metrics on Calabi-Yau cones and develops a new $C^{ ext{alpha}}$-estimate for bounded K"ahler metrics with scalar curvature bounds.

Findings

Any cscK metric with bounded ratio to the cone metric is equivalent to it up to automorphism.

Established a $C^{0, ext{alpha}}$-estimate for bounded K"ahler metrics near the apex.

Proved such metrics are asymptotic to the Ricci-flat cone with polynomial decay.

Abstract

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