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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)"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1601.02959","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-01-12T17:03:03Z","cross_cats_sorted":[],"title_canon_sha256":"ae36a60bba0dfcefb35f64a8f3adfffc323a64e93e147658d24da3aba622a93e","abstract_canon_sha256":"6d3c49790a8aaaf028e5a8d6f70a4c4b9f8b51cfe9de455b30a2e628ff4c15b3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:59.243365Z","signature_b64":"ISa8xc0R4/YEQe7YZKdIZEUL6Ft15NJOe0tgkr4itW8Xr6zWAn43EyZAmzkoutgxP4Fn+uvPWOnaQXCSn8bMAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6679f9f9f21855df5194244187625eaea6c9f5034c1d1becb171e64ab5d4fdf7","last_reissued_at":"2026-05-18T01:22:59.242502Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:59.242502Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1601.02959","created_at":"2026-05-18T01:22:59.242631+00:00"},{"alias_kind":"arxiv_version","alias_value":"1601.02959v1","created_at":"2026-05-18T01:22:59.242631+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.02959","created_at":"2026-05-18T01:22:59.242631+00:00"},{"alias_kind":"pith_short_12","alias_value":"MZ47T6PSDBK5","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_16","alias_value":"MZ47T6PSDBK56UMU","created_at":"2026-05-18T12:30:32.724797+00:00"},{"alias_kind":"pith_short_8","alias_value":"MZ47T6PS","created_at":"2026-05-18T12:30:32.724797+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MZ47T6PSDBK56UMUERAYOYS6V2","json":"https://pith.science/pith/MZ47T6PSDBK56UMUERAYOYS6V2.json","graph_json":"https://pith.science/api/pith-number/MZ47T6PSDBK56UMUERAYOYS6V2/graph.json","events_json":"https://pith.science/api/pith-number/MZ47T6PSDBK56UMUERAYOYS6V2/events.json","paper":"https://pith.science/paper/MZ47T6PS"},"agent_actions":{"view_html":"https://pith.science/pith/MZ47T6PSDBK56UMUERAYOYS6V2","download_json":"https://pith.science/pith/MZ47T6PSDBK56UMUERAYOYS6V2.json","view_paper":"https://pith.science/paper/MZ47T6PS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1601.02959&json=true","fetch_graph":"https://pith.science/api/pith-number/MZ47T6PSDBK56UMUERAYOYS6V2/graph.json","fetch_events":"https://pith.science/api/pith-number/MZ47T6PSDBK56UMUERAYOYS6V2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MZ47T6PSDBK56UMUERAYOYS6V2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MZ47T6PSDBK56UMUERAYOYS6V2/action/storage_attestation","attest_author":"https://pith.science/pith/MZ47T6PSDBK56UMUERAYOYS6V2/action/author_attestation","sign_citation":"https://pith.science/pith/MZ47T6PSDBK56UMUERAYOYS6V2/action/citation_signature","submit_replication":"https://pith.science/pith/MZ47T6PSDBK56UMUERAYOYS6V2/action/replication_record"}},"created_at":"2026-05-18T01:22:59.242631+00:00","updated_at":"2026-05-18T01:22:59.242631+00:00"}