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When the Lagrange function ${\\cal E}$ is a polynomial of degree $n$ of the mean curvature $H$ of the surface, the radii ($a,r$) of the 2-torus are related as $\\frac{a^2}{r^2}=\\frac{n^2-n}{n^2-n-1}, n \\ge 2$. If the Lagrange function depends on both mean and Gaussian curvatures, the 2- torus remains to be a critical point of ${\\cal F}_{n}$ without any constraints on the radii of the torus."},"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":"1401.7192","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-01-28T14:18:00Z","cross_cats_sorted":["hep-th","math-ph","math.MP"],"title_canon_sha256":"27bc400ef397b36001924ee284ba3c63ba6b66b8ae4cb0defe997b5b7569578f","abstract_canon_sha256":"0ad7bce915c992268a81c096b29480d7574b24741a426b0e0b2c214a25013dcc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:00:38.887062Z","signature_b64":"ZZL/WizVCWzTqtYo9Tarp7ayDf+F9RgmaUnSnOCMy5hwyVThaJ77JkNDgNr49ddsyn2fVX0eZjuJ/XT58Z1lDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5bba96e65888f1fa51f1064588c8e03899fab58fe965bf3ec2a6e4a3b9ee686","last_reissued_at":"2026-05-18T03:00:38.886346Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:00:38.886346Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Functionals on Closed 2-Surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math-ph","math.MP"],"primary_cat":"math.DG","authors_text":"Metin Gurses","submitted_at":"2014-01-28T14:18:00Z","abstract_excerpt":"We show that the 2-torus in ${\\mathbb R}^3$ is a critical point of a sequence of functionals ${\\cal F}_{n}$ ($n=1,2,3, \\cdots$) defined over compact 2-surfaces in ${\\mathbb R}^3$. 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