{"paper":{"title":"Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure","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","submitted_at":"2016-05-09T13:55:12Z","abstract_excerpt":"Let $\\mathbb{M}$ be a compact $C^\\infty$-smooth Riemannian manifold of dimension $n$, $n\\geq 3$, and let $\\varphi_\\lambda: \\Delta_M \\varphi_\\lambda + \\lambda \\varphi_\\lambda = 0$ denote the Laplace eigenfunction on $\\mathbb{M}$ corresponding to the eigenvalue $\\lambda$. We show that $$H^{n-1}(\\{ \\varphi_\\lambda=0\\}) \\leq C \\lambda^{\\alpha},$$ where $\\alpha>1/2$ is a constant, which depends on $n$ only, and $C>0$ depends on $\\mathbb{M}$ . This result is a consequence of our study of zero sets of harmonic functions on $C^\\infty$-smooth Riemannian manifolds. We develop a technique of propagation "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.02587","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"}