{"paper":{"title":"Morse theory for Lagrange multipliers and adiabatic limits","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.SG"],"primary_cat":"math.GT","authors_text":"Guangbo Xu, Stephen Schecter","submitted_at":"2012-11-13T15:54:08Z","abstract_excerpt":"Given two Morse functions $f, \\mu$ on a compact manifold $M$, we study the Morse homology for the Lagrange multiplier function on $M \\times {\\mathbb R}$ which sends $(x, \\eta)$ to $f(x) + \\eta \\mu(x)$. Take a product metric on $M \\times {\\mathbb R}$, and rescale its ${\\mathbb R}$-component by a factor $\\lambda^2$. We show that generically, for large $\\lambda$, the Morse-Smale-Witten chain complex is isomorphic to the one for $f$ and the metric restricted to ${\\mu^{-1}(0)}$, with grading shifted by one. On the other hand, let $\\lambda\\to 0$, we obtain another chain complex, which is geometrical"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.3028","kind":"arxiv","version":3},"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"}