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This completely characterizes the equality in Zhong-Yang type sharp spectral gap estimates in the metric measure setting with Riemannian lower Ricci bounds. Among such spaces, are the familiar Riemannian manifolds with $\\Ric \\ge 0$, $(0,N)-$ Bakry-\\'{E}mery manifolds, $(0,n)-"},"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":"1506.04936","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-06-16T12:20:42Z","cross_cats_sorted":[],"title_canon_sha256":"8758321aecf297660d25eefc8229c8de3ace66e139c00e6d2d3188732c59bc44","abstract_canon_sha256":"a601b9c73a51a14ad35345b9154fbc550f0f184ab66649f0408bb8d2be68f05a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:43.504583Z","signature_b64":"IO9G2rMhaIdSgHJDfRSEJSqq6RIAZY8Al1q9yuvbV/kKYjYERIC0bB6UwzYUYkyjMUw/4QydEn9Q/M/kJoAiAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"74c119707af59aa76c9ea29aad00f8b93e44d9026e4d87ab283e2f725f0cf457","last_reissued_at":"2026-05-17T23:55:43.504039Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:43.504039Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Characterization of equality in Zhong-Yang type (sharp) spectral gap estimates for metric measure spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Sajjad Lakzian","submitted_at":"2015-06-16T12:20:42Z","abstract_excerpt":"We prove that a compact $RCD^*(0,N)$ (or equivalently $RCD(0,N)$) metric measure space, $\\left(X, d, m \\right)$, with $\\diam X \\le d$ and its first (nonzero) eigenvalue of the Laplacian (in the sense of Ambrosio-Gigli-Savar\\'{e}) , $\\lambda_1 = \\frac{\\pi^2}{d^2}$, has to be a circle or a line segment with diameter, $\\pi$. This completely characterizes the equality in Zhong-Yang type sharp spectral gap estimates in the metric measure setting with Riemannian lower Ricci bounds. 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