{"paper":{"title":"The Cartan-Hadamard conjecture and The Little Prince","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Beno\\^it Kloeckner (LAMA, Greg Kuperberg (UC Davis), If)","submitted_at":"2013-03-13T09:45:13Z","abstract_excerpt":"The generalized Cartan-Hadamard conjecture says that if $\\Omega$ is a domain with fixed volume in a complete, simply connected Riemannian $n$-manifold $M$ with sectional curvature $K \\le \\kappa \\le 0$, then the boundary of $\\Omega$ has the least possible boundary volume when $\\Omega$ is a round $n$-ball with constant curvature $K=\\kappa$.  The case $n=2$ and $\\kappa=0$ is an old result of Weil.  We give a unified proof of this conjecture in dimensions $n=2$ and $n=4$ when $\\kappa=0$, and a special case of the conjecture for $\\kappa \\textless{} 0$ and a version for $\\kappa \\textgreater{} 0$.  O"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.3115","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"}