{"paper":{"title":"The inhomogeneous Allen--Cahn equation and the existence of prescribed-mean-curvature hypersurfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Costante Bellettini, Neshan Wickramasekera","submitted_at":"2020-10-12T16:51:54Z","abstract_excerpt":"We prove that for any given compact Riemannian manifold $N$ of dimension $n+1 \\geq 3$ and any non-negative Lipschitz function $g$ on $N$, there exists a quasi-embedded, boundaryless hypersurface $M \\subset N,$ of class $C^{2, \\alpha}$ for any $\\alpha \\in (0,1),$ such that $M$ is the image of a two-sided immersion whose mean curvature is given by $g\\nu$ for an appropriate choice of continuous unit normal $\\nu$ to the immersion; and moreover, the singular set $\\Sigma = \\overline{M} \\setminus M$ is empty if $2 \\leq n \\leq 6,$ finite if $n=7$ and satisfies ${\\mathcal H}^{n-7 + \\gamma}(\\Sigma) = 0$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2010.05847","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2010.05847/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}