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Einstein metrics are critical points of the total scalar curvature functional for fixed volume. The scalar curvature $S$ of a left-invariant metric $g$ is constant and can be expressed as a rational function in the parameters determining the metric. The critical points of $S$, subject to the volume constraint, are given by the zero locus of a system of pol"},"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":"1703.10512","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-03-30T15:10:03Z","cross_cats_sorted":["hep-th","math-ph","math.MP"],"title_canon_sha256":"a9182fde9ba065c4736627c2800bab03825908dfc140e07458645588f7450954","abstract_canon_sha256":"cb1983dc0797b51e3cc21d5ebd055322c67b34db9e5d8410339dd202f75fb1cf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:11:18.980332Z","signature_b64":"LeU59g0/4B7DPyl8BCMuBjoK6pBDVZDG88MClxpiW+8cZOWFT9UHDXZ5BUzr+z3RHd/OCLlPeTIftitf5HRSCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6f5053ec58e891a12e805c9743880661202e58f8d5cb44e5fccc0b754f83908f","last_reissued_at":"2026-05-18T00:11:18.979678Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:11:18.979678Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Left-invariant Einstein metrics on $S^3 \\times S^3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math-ph","math.MP"],"primary_cat":"math.DG","authors_text":"Alexander S. Haupt, David Lindemann, Florin Belgun, Vicente Cort\\'es","submitted_at":"2017-03-30T15:10:03Z","abstract_excerpt":"The classification of homogeneous compact Einstein manifolds in dimension six is an open problem. We consider the remaining open case, namely left-invariant Einstein metrics $g$ on $G = \\mathrm{SU}(2) \\times \\mathrm{SU}(2) = S^3 \\times S^3$. Einstein metrics are critical points of the total scalar curvature functional for fixed volume. The scalar curvature $S$ of a left-invariant metric $g$ is constant and can be expressed as a rational function in the parameters determining the metric. 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