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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.02361 |
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| _version_ | 1866913678022934528 |
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| author | Klemmensen, Johan Jacoby |
| author_facet | Klemmensen, Johan Jacoby |
| contents | 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 $α> 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_02361 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Liouville Theorem and $C^α$-Estimate for Calabi-Yau Cones Klemmensen, Johan Jacoby Differential Geometry Complex Variables 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 $α> 0$. |
| title | A Liouville Theorem and $C^α$-Estimate for Calabi-Yau Cones |
| topic | Differential Geometry Complex Variables |
| url | https://arxiv.org/abs/2502.02361 |