Globally subanalytic CMC surfaces in \mathbb{R}^3

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2014, 21: 186-192. doi: 10.3934/era.2014.21.186

Globally subanalytic CMC surfaces in $\mathbb{R}^3$

J. L. Barbosa ^1, , L. Birbrair ^2, , M. do Carmo ^3, and A. Fernandes ^4,

1.	Rua Carolina Sucupira 723 ap 2002, 60140-120, Fortaleza-CE, Brazil
2.	Departamento de Matemática, Universidade Federal do Ceará Av. Hum- berto Monte, s/n Campus do Pici - Bloco 914, 60455-760, Fortaleza-CE, Brazil
3.	Instituto Nacional de Matemática Pura e Aplicada, IMPA Estrada Dona Castorina 110, 22460-320, Rio de Janeiro-RJ, Brazil
4.	Departamento de Matemática, Universidade Federal do Ceará Av. Humberto Monte, s/n Campus do Pici - Bloco 914, 60455-760, Fortaleza-CE, Brazil

Received June 2014 Published December 2014

Abstract
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We prove that globally subanalytic nonsingular CMC surfaces of $\mathbb{R}^3$ are only planes, round spheres, or right circular cylinders.

Keywords: Globally Subanalytic Sets., CMC surfaces.

Mathematics Subject Classification: 53C42; 53A1.

Citation: J. L. Barbosa, L. Birbrair, M. do Carmo, A. Fernandes. Globally subanalytic CMC surfaces in $\mathbb{R}^3$. Electronic Research Announcements, 2014, 21: 186-192. doi: 10.3934/era.2014.21.186

References:

[1]	A. Alexandrov, A characteristic property of spheres, Ann. Mat. Pura Appl., 58 (1962), 303-315. doi: 10.1007/BF02413056.
[2]	J. Barbosa and M. do Carmo, On regular algebraic surfaces of $\mathbb{R}^3$ with constant mean curvature, arXiv:1403.7029, (2014).
[3]	E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math., 67 (1988), 5-42.
[4]	M. Coste, An Introduction to Semialgebraic Geometry, Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligrafici Internazionali, Pisa, 2000. Available from: http://perso.univ-rennes1.fr/michel.coste/articles.html.
[5]	M. Coste, An Introduction to O-minimal Geometry, Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligrafici Internazionali, Pisa, 2000. Available from: http://perso.univ-rennes1.fr/michel.coste/articles.html.
[6]	L. van den Dries, A generalization of the Tarski-Seidenberg theorem, and some nondefinability results, Bulletin Amer. Math. Soc. (N.S.), 15 (1986), 189-193. doi: 10.1090/S0273-0979-1986-15468-6.
[7]	L. van den Dries and C. Miller, Geometric categories and o-minimal structures, Duke Math. J., 84 (1996), 467-540. doi: 10.1215/S0012-7094-96-08416-1.
[8]	A. Gabrièlov, Projections of semianalytic sets, Funkcional. Anal. i Priložen., 2 (1968), 18-30.
[9]	H. Hopf, Über Flächen mit einer Relation zwischen Hauptkrümmungen, Math Nachr., 4 (1951), 232-249.
[10]	D. Hoffman and W. H. Meeks, III, The strong halfspace theorem for minimal surfaces, Invent. Math., 101 (1990), 373-377. doi: 10.1007/BF01231506.
[11]	N. Korevaar, R. Kusner and B. Solomon, The struture of complete embedded surfaces with constant mean curvature, J. Differential Geom., 30 (1989), 465-503.
[12]	S. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. Pisa (3), 18 (1964), 449-474.
[13]	R. Osserman, Global properties of minimal surfaces in $E^3$ and $E^n$, Ann. of Math. (2), 80 (1964), 340-364. doi: 10.2307/1970396.
[14]	R. Schoen, Uniqueness, symmetry, and embeddedness of minimal surfaces, J. Differential Geom., 18 (1983), 791-809.
[15]	A. Tarski, A Decision Method for an Elementary Algebra and Geometry, 2nd edition, University of California Press, Berkeley and Los Angeles, Calif., 1951.

show all references

References:

[1]	A. Alexandrov, A characteristic property of spheres, Ann. Mat. Pura Appl., 58 (1962), 303-315. doi: 10.1007/BF02413056.
[2]	J. Barbosa and M. do Carmo, On regular algebraic surfaces of $\mathbb{R}^3$ with constant mean curvature, arXiv:1403.7029, (2014).
[3]	E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math., 67 (1988), 5-42.
[4]	M. Coste, An Introduction to Semialgebraic Geometry, Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligrafici Internazionali, Pisa, 2000. Available from: http://perso.univ-rennes1.fr/michel.coste/articles.html.
[5]	M. Coste, An Introduction to O-minimal Geometry, Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligrafici Internazionali, Pisa, 2000. Available from: http://perso.univ-rennes1.fr/michel.coste/articles.html.
[6]	L. van den Dries, A generalization of the Tarski-Seidenberg theorem, and some nondefinability results, Bulletin Amer. Math. Soc. (N.S.), 15 (1986), 189-193. doi: 10.1090/S0273-0979-1986-15468-6.
[7]	L. van den Dries and C. Miller, Geometric categories and o-minimal structures, Duke Math. J., 84 (1996), 467-540. doi: 10.1215/S0012-7094-96-08416-1.
[8]	A. Gabrièlov, Projections of semianalytic sets, Funkcional. Anal. i Priložen., 2 (1968), 18-30.
[9]	H. Hopf, Über Flächen mit einer Relation zwischen Hauptkrümmungen, Math Nachr., 4 (1951), 232-249.
[10]	D. Hoffman and W. H. Meeks, III, The strong halfspace theorem for minimal surfaces, Invent. Math., 101 (1990), 373-377. doi: 10.1007/BF01231506.
[11]	N. Korevaar, R. Kusner and B. Solomon, The struture of complete embedded surfaces with constant mean curvature, J. Differential Geom., 30 (1989), 465-503.
[12]	S. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. Pisa (3), 18 (1964), 449-474.
[13]	R. Osserman, Global properties of minimal surfaces in $E^3$ and $E^n$, Ann. of Math. (2), 80 (1964), 340-364. doi: 10.2307/1970396.
[14]	R. Schoen, Uniqueness, symmetry, and embeddedness of minimal surfaces, J. Differential Geom., 18 (1983), 791-809.
[15]	A. Tarski, A Decision Method for an Elementary Algebra and Geometry, 2nd edition, University of California Press, Berkeley and Los Angeles, Calif., 1951.

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