{"paper":{"title":"The weighted $\\sigma_k$-curvature of a smooth metric measure space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Jeffrey S. Case","submitted_at":"2016-08-04T19:56:10Z","abstract_excerpt":"We propose a definition of the weighted $\\sigma_k$-curvature of a smooth metric measure space and justify it in two ways. First, we show that the weighted $\\sigma_k$-curvature prescription problem is governed by a fully nonlinear second order elliptic PDE which is variational when $k=1,2$ or the smooth metric measure space is locally conformally flat in the weighted sense. Second, we show that, in the variational cases, quasi-Einstein metrics are stable with respect to the total weighted $\\sigma_k$-curvature functional. We also discuss related conjectures for weighted Einstein manifolds."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.01663","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"}