{"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 ${\\mathbb R}^3$. 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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.7192","kind":"arxiv","version":2},"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"}