Abstract
Let \(x:{\text M}^{n}\rightarrow\,{\text R}^{n+1}\) be an immersion of an orientable, n-dimensional manifold \({\text M}^{n}\) into the euclidean space \({\text R}^{n+1}\). The condition that x has nonzero constant mean curvature H = H 0 is known to be equivalent to the fact that xis a critical point of a variational problem.
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References
Barbosa, J.L., do Carmo, M.: Stability of minimal surfaces and eigenvalues of the Laplacian. Math. Z. 173 13- 28 (1980)
Böhme, R., Tomi, F.: Zur Struktur der Lösungsmenge des Plateauproblems. Math. Z. 133, 1-29 (1973)
Bolza, 0.: Vorlesungen über Variationsrechnung, Berlin-Leipzig: Teubner 1909
do Carmo, M., Peng, C.K.: Stable complete minimal surfaces in R3 are planes. Bull. Amer. Math. Soc. (N.S.) 1, 903-906 (1979)
Fischer-Colbrie, D., Schoen, R.: The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature. Comm. Pure Appl. Math. 35, 199-211 (1980)
Frid, H.: 0 Teorema do ìndice de Morse. Master's thesis, IMPA 1982
Frid, H., Thayer, F.J.: The Morse index theorem for elliptic problems with a constraint. An. Acad. Brasil. Ci. 55, 9-10 (1983)
Hsiang, W.Y., Teng, Z.H., Yu, W.: New examples of constant mean curvature immersions of (2k-1)-spheres into euclidean 2k-space. Ann. of Math. 117, 609- 625 (1983)
Kenmotsu, K.: Surfaces of revolution with prescribed mean curvature. Tôhoku Math. J. 32 , 147-153 (1980)
Lawson, B., Jr.: Lectures on Minimal Submanifolds, vol. 1. Berkeley: Publish or Perish 1980
Mori, H.: Stable constant mean curvature surfaces in R 3 and in H3. Preprint
Pogorelov, A.V.: On the stability of minimal surfaces. Soviet Math. Dokl. 24, 274-276 (1981)
Ruchert, H.: Ein Eindeutigkeitssatz für Flächen konstanter mittlerer Krümmung. Arch. Math. (Basel) 33, 91-104 (1979)
D'Arcy Thompson: On growth and form. New York: Cambridge at the University Press and The MacMillan Co 1945
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Barbosa, J.L., Carmo, M.d. (2012). Stability of Hypersurfaces with Constant Mean Curvature. In: Tenenblat, K. (eds) Manfredo P. do Carmo – Selected Papers. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25588-5_18
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