{"paper":{"title":"Energy Concentration for Min-Max Solutions of the Ginzburg-Landau Equations on Manifolds with $b_1(M)\\neq 0$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Daniel Stern","submitted_at":"2017-04-03T17:54:15Z","abstract_excerpt":"We establish a new estimate for the Ginzburg-Landau energies $E_{\\epsilon}(u)=\\int_M\\frac{1}{2}|du|^2+\\frac{1}{4\\epsilon^2}(1-|u|^2)^2$ of complex-valued maps $u$ on a compact, oriented manifold $M$ with $b_1(M)\\neq 0$, obtained by decomposing the harmonic component $h_u$ of the one-form $ju:=u^1du^2-u^2du^1$ into an integral and fractional part. We employ this estimate to show that, for critical points $u_{\\epsilon}$ of $E_{\\epsilon}$ arising from the two-parameter min-max construction considered by the author in previous work, a nontrivial portion of the energy must concentrate on a stationa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.00712","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"}