{"paper":{"title":"Hessian curvature hypersurfaces with prescribed Gauss image","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Huang Rongli, Qu Changzheng, Wang ZhiZhang, Wo Weifeng","submitted_at":"2024-12-12T10:44:39Z","abstract_excerpt":"In this paper, we investigate Hessian curvature hypersurfaces with prescribed Gauss images. Given geodesically strictly convex bounded domains $\\Omega$ in $\\mathbb{R}^n$ and $\\tilde{\\Omega}$ in the unit hemisphere, we prove that there is a strictly convex graphic hypersurface defined in $\\Omega$ with prescribed $k$-Hessian curvatures such that its Gauss image is $\\tilde{\\Omega}$. Our proof relies on a novel $C^2$ boundary estimate which utilizes the orthogonal invariance of hypersurfaces. Indeed, we employ some special vector fields generated by the infinitesimal rotations in $\\mathbb{R}^{n+1}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.09159","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/2412.09159/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"}