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

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 $α> 0$.