{"paper":{"title":"Cartan's and Gauss's equations and rigidity theorems for isometric embeddings in low Sobolev regularity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Isaac Newell, Luc Nguyen","submitted_at":"2026-07-02T16:44:12Z","abstract_excerpt":"Let $\\{\\eta^i\\}_{i=1}^2$ be a an orthonormal coframe on a domain $U$ on a smooth surface $(\\Sigma,g)$. When $\\eta^i$ is smooth, it is well-known that there is a unique connection 1-form $\\omega$ verifying Cartan's first structural equations $d\\eta^i = (*\\eta^i) \\wedge \\omega$, and Cartan's second structural equation $d\\omega = K_g dvol_g$. We prove that this statement remains valid when the frame is $C^0 \\cap H^{\\frac12}$, where the structural equations are understood in the sense of distributions. From this, we deduce that the Gauss equation $\\mathrm{Det}\\, D^2 f = K_g (1+|Df|^2)^2$ holds for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02412","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/2607.02412/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"}