{"paper":{"title":"Symmetries in some extremal problems between two parallel hyperplanes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Monica Moulin Ribeiro Merkle","submitted_at":"2016-01-12T17:03:03Z","abstract_excerpt":"Let $M$ be a compact hypersurface with boundary $\\partial M=\\partial D_1 \\cup \\partial D_2$, $\\partial D_1 \\subset \\Pi _1$, $\\partial D_2 \\subset \\Pi _2$, $\\Pi_1$ and $\\Pi _2$ two parallel hyperplanes in $\\mathbb{R}^{n+1}$ ($n \\geq 2$). Suppose that $M$ is contained in the slab determined by these hyperplanes and that the mean curvature $H$ of $M$ depends only on the distance $u$ to $\\Pi _i$, $i=1,2$. We prove that these hypersurfaces are symmetric to a perpendicular orthogonal to $\\Pi _i$, $i=1,2$, under different conditions imposed on the boundary of hypersurfaces on the parallel planes: (i)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.02959","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"}