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

Let $(\mathcal{C},{\omega }_{\mathcal{C}})$ be a Ricci-flat, simply connected, conical Kähler manifold. We establish a Liouville theorem for constant scalar curvature Kähler (cscK) metrics on $\mathcal{C}$. The theorem asserts that any cscK metric $\omega$ satisfying the uniform bound $\frac{1}{C}{\omega }_{\mathcal{C}}\le \omega \le C{\omega }_{\mathcal{C}}$ for some $C\ge 1$ is equal to ${\omega }_{\mathcal{C}}$ 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 Kähler metrics on a ball around the apex, using a Hölder-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 Kähler metric $\omega$ is asymptotic to the Ricci-flat cone metric ${\omega }_{\mathcal{C}}$, with polynomial decay rate $r^{\alpha }$ and for sufficiently small $\alpha >0$.