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The upper bound is given in terms of an integration involving $tr T$ and $|H_T|^2$, where $tr T$ is the trace of the tensor $T$ and $H_T=\\sum_{i=1}^nA(Te_i,e_i)$ is a normal vector field associated with $T$ and the second fundamental form $A$ of"},"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":"1806.10826","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-06-28T08:32:49Z","cross_cats_sorted":[],"title_canon_sha256":"1fd505d9600defba5bc01fa161a9af62816e3610c8450cf8b49549fda258c429","abstract_canon_sha256":"38885ec91b14f21a93dd3f57d984fbd48cccdffe652137154c8582071d7e1e13"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:07.818538Z","signature_b64":"igcz7Hgtnr7hlec4f5eGP2p3ro6w0AwuH6lqKgdnPi9eWrC61NlKpEQNacaDwTQ2Cjht9OIHpgDU+FfjMJPQBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9c04370ab55b25fb31feacb73d3a77b69dcfc5e2e9ad0f5bd79411e6f7705340","last_reissued_at":"2026-05-18T00:12:07.817873Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:07.817873Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp Reilly-type inequalities for submanifolds in space forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hang Chen, Xianfeng Wang","submitted_at":"2018-06-28T08:32:49Z","abstract_excerpt":"Let $M$ be an $n(>2)$-dimensional closed orientable submanifold in an $(n+p)$-dimensional space form $\\mathbb{R}^{n+p}(c)$. 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