{"paper":{"title":"Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimension two and three","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.DG","math.SP"],"primary_cat":"math.AP","authors_text":"Alexander Logunov, Eugenia Malinnikova","submitted_at":"2016-05-09T14:09:23Z","abstract_excerpt":"Let $\\Delta_M$ be the Laplace operator on a compact $n$-dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions $u:\\Delta u + \\lambda u =0$. In dimension $n=2$ we refine the Donnelly-Fefferman estimate by showing that $H^1(\\{u=0 \\})\\le C\\lambda^{3/4-\\beta}$, $\\beta \\in (0,1/4)$. The proof employs the Donnelli-Fefferman estimate and a combinatorial argument, which also gives a lower (non-sharp) bound in dimension $n=3$: $H^2(\\{u=0\\})\\ge c\\lambda^\\alpha$, $\\alpha \\in (0,1/2)$. The positive constants $c,C$ depend on the manifold, $\\alpha$ and $\\beta$ are uni"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.02595","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"}