{"paper":{"title":"Minimizing the Gauss map area of surfaces in $\\mathbb{S}^3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Gerard Orriols, Tristan Rivi\\`ere","submitted_at":"2025-06-05T15:22:55Z","abstract_excerpt":"We establish the lower bound of $4\\pi(1+g)$ for the area of the Gauss map of any immersion of a closed oriented surface of genus $g$ into $\\mathbb{S}^3$, taking values in the Grassmannian of $2$-planes in $\\mathbb{R}^4$. This lower bound is proved to be optimal for any genus $g \\in \\mathbb{N}$ but attained only when $g = 0$. For $g \\neq 0$ we describe the behavior of any minimizing sequence of embeddings: we prove that, modulo extraction of a subsequence, the surfaces converge in the Hausdorff distance to a round sphere $S$, and the integral cycles carried by the Gauss maps split into $g+1$ sp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.05141","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.05141/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}