{"paper":{"title":"Complex Gradient Systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CV","authors_text":"Giuseppe Tomassini, Sergio Venturini","submitted_at":"2011-06-28T13:55:18Z","abstract_excerpt":"Let $M$ be a complex manifold of complex dimension $n+k$. We say that the functions $u_1,...s,u_k$ and the vector fields $\\xi_1,...,\\xi_k$ on $M$ form a \\emph{complex gradient system} if $\\xi_1,...,\\xi_k,J\\xi_1,...,J\\xi_k$ are linearly independent at each point $p\\in M$ and generate an integrable distribution of $TM$ of dimension $2k$ and $du_\\alpha(\\xi_\\beta)=0$, $\\d^c\\u_\\alpha(\\xi_\\beta)=\\delta_{\\alpha\\beta}$ for $\\alpha,\\beta=1,...,k$. We prove a Cauchy theorem for such complex gradient systems with initial data along a $\\CR-$submanifold of type $(\\CRdim,\\CRcodim)$. We also give a complete "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.5666","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"}