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