{"paper":{"title":"The mean curvature flow along the K\\\"ahler-Ricci flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jiayu Li, Xiaoli Han","submitted_at":"2011-05-06T01:06:41Z","abstract_excerpt":"Let $(M,\\overline{g})$ be a K\\\"ahler surface, and $\\Sigma$ an immersed surface in $M$. The K\\\"ahler angle of $\\Sigma$ in $M$ is introduced by Chern-Wolfson \\cite{CW}. Let $(M,\\overline{g}(t))$ evolve along the K\\\"ahler-Ricci flow, and $\\Sigma_t$ in $(M,\\overline{g}(t))$ evolve along the mean curvature flow. We show that the K\\\"ahler angle $\\alpha(t)$ satisfies the evolution equation: $$ (\\frac{\\partial}{\\partial t}-\\Delta)\\cos\\alpha=|\\overline\\nabla J_{\\Sigma_t}|^2\\cos\\alpha+R\\sin^2\\alpha\\cos\\alpha, $$ where $R$ is the scalar curvature of $(M, \\overline{g}(t))$.\n  The equation implies that, if"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.1200","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"}