{"paper":{"title":"Toric Surfaces, K-Stability and Calabi Flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hongnian Huang","submitted_at":"2012-07-25T12:13:38Z","abstract_excerpt":"Let $X$ be a toric surface and $u$ be a normalized symplectic potential on the corresponding polygon $P$. Suppose that the Riemannian curvature is bounded by a constant $C_1$ and $\\int_{\\partial P} u ~ d \\sigma < C_2, $ then there exists a constant $C_3$ depending only on $C_1, C_2$ and $P$ such that the diameter of $X$ is bounded by $C_3$. Moreoever, we can show that there is a constant $M > 0$ depending only on $C_1, C_2$ and $P$ such that Donaldson's $M$-condition holds for $u$. As an application, we show that if $(X,P)$ is (analytic) relative $K$-stable, then the modified Calabi flow conve"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.5964","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}