{"paper":{"title":"Quantitative Stratification and the Regularity of Mean Curvature Flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Aaron Naber, Jeff Cheeger, Robert Haslhofer","submitted_at":"2012-07-16T10:14:32Z","abstract_excerpt":"Let $\\cM$ be a Brakke flow of $n$-dimensional surfaces in $R^N$. The singular set $\\cS\\subset\\cM$ has a stratification $\\cS^0\\subset\\cS^1\\subset...\\cS$, where $X\\in \\cS^j$ if no tangent flow at $X$ has more than $j$ symmetries. Here, we define quantitative singular strata $\\cS^j_{\\eta,r}$ satisfying $\\cup_{\\eta>0}\\cap_{0<r} \\cS^j_{\\eta,r}=\\cS^j$. Sharpening the known parabolic Hausdorff dimension bound $\\dim \\cS^j\\leq j$, we prove the effective Minkowski estimates that the volume of $r$-tubular neighborhoods of $\\cS^j_{\\eta,r}$ satisfies $\\Vol (T_r(\\cS^j_{\\eta,r})\\cap B_1)\\leq Cr^{N+2-j-\\varep"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.3619","kind":"arxiv","version":2},"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"}