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

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

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

We prove that globally subanalytic nonsingular CMC surfaces of $\mathbb{R}^3$ are only planes, round spheres, or right circular cylinders.
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,, \emph{Ann. Mat. Pura Appl.}, 58 (1962), 303.  doi: 10.1007/BF02413056.  Google Scholar

[2]

J. Barbosa and M. do Carmo, On regular algebraic surfaces of $\mathbbR^3$ with constant mean curvature,, \arXiv{1403.7029}, (2014).   Google Scholar

[3]

E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets,, \emph{Inst. Hautes Études Sci. Publ. Math.}, 67 (1988), 5.   Google Scholar

[4]

M. Coste, An Introduction to Semialgebraic Geometry,, Dip. Mat. Univ. Pisa, (2000).   Google Scholar

[5]

M. Coste, An Introduction to O-minimal Geometry,, Dip. Mat. Univ. Pisa, (2000).   Google Scholar

[6]

L. van den Dries, A generalization of the Tarski-Seidenberg theorem, and some nondefinability results,, \emph{Bulletin Amer. Math. Soc. (N.S.)}, 15 (1986), 189.  doi: 10.1090/S0273-0979-1986-15468-6.  Google Scholar

[7]

L. van den Dries and C. Miller, Geometric categories and o-minimal structures,, \emph{Duke Math. J.}, 84 (1996), 467.  doi: 10.1215/S0012-7094-96-08416-1.  Google Scholar

[8]

A. Gabrièlov, Projections of semianalytic sets,, \emph{Funkcional. Anal. i Priložen.}, 2 (1968), 18.   Google Scholar

[9]

H. Hopf, Über Flächen mit einer Relation zwischen Hauptkrümmungen,, \emph{Math Nachr.}, 4 (1951), 232.   Google Scholar

[10]

D. Hoffman and W. H. Meeks, III, The strong halfspace theorem for minimal surfaces,, \emph{Invent. Math.}, 101 (1990), 373.  doi: 10.1007/BF01231506.  Google Scholar

[11]

N. Korevaar, R. Kusner and B. Solomon, The struture of complete embedded surfaces with constant mean curvature,, \emph{J. Differential Geom.}, 30 (1989), 465.   Google Scholar

[12]

S. Lojasiewicz, Triangulation of semi-analytic sets,, \emph{Ann. Scuola Norm. Sup. Pisa (3)}, 18 (1964), 449.   Google Scholar

[13]

R. Osserman, Global properties of minimal surfaces in $E^3$ and $E^n$,, \emph{Ann. of Math. (2)}, 80 (1964), 340.  doi: 10.2307/1970396.  Google Scholar

[14]

R. Schoen, Uniqueness, symmetry, and embeddedness of minimal surfaces,, \emph{J. Differential Geom.}, 18 (1983), 791.   Google Scholar

[15]

A. Tarski, A Decision Method for an Elementary Algebra and Geometry,, 2nd edition, (1951).   Google Scholar

show all references

References:
[1]

A. Alexandrov, A characteristic property of spheres,, \emph{Ann. Mat. Pura Appl.}, 58 (1962), 303.  doi: 10.1007/BF02413056.  Google Scholar

[2]

J. Barbosa and M. do Carmo, On regular algebraic surfaces of $\mathbbR^3$ with constant mean curvature,, \arXiv{1403.7029}, (2014).   Google Scholar

[3]

E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets,, \emph{Inst. Hautes Études Sci. Publ. Math.}, 67 (1988), 5.   Google Scholar

[4]

M. Coste, An Introduction to Semialgebraic Geometry,, Dip. Mat. Univ. Pisa, (2000).   Google Scholar

[5]

M. Coste, An Introduction to O-minimal Geometry,, Dip. Mat. Univ. Pisa, (2000).   Google Scholar

[6]

L. van den Dries, A generalization of the Tarski-Seidenberg theorem, and some nondefinability results,, \emph{Bulletin Amer. Math. Soc. (N.S.)}, 15 (1986), 189.  doi: 10.1090/S0273-0979-1986-15468-6.  Google Scholar

[7]

L. van den Dries and C. Miller, Geometric categories and o-minimal structures,, \emph{Duke Math. J.}, 84 (1996), 467.  doi: 10.1215/S0012-7094-96-08416-1.  Google Scholar

[8]

A. Gabrièlov, Projections of semianalytic sets,, \emph{Funkcional. Anal. i Priložen.}, 2 (1968), 18.   Google Scholar

[9]

H. Hopf, Über Flächen mit einer Relation zwischen Hauptkrümmungen,, \emph{Math Nachr.}, 4 (1951), 232.   Google Scholar

[10]

D. Hoffman and W. H. Meeks, III, The strong halfspace theorem for minimal surfaces,, \emph{Invent. Math.}, 101 (1990), 373.  doi: 10.1007/BF01231506.  Google Scholar

[11]

N. Korevaar, R. Kusner and B. Solomon, The struture of complete embedded surfaces with constant mean curvature,, \emph{J. Differential Geom.}, 30 (1989), 465.   Google Scholar

[12]

S. Lojasiewicz, Triangulation of semi-analytic sets,, \emph{Ann. Scuola Norm. Sup. Pisa (3)}, 18 (1964), 449.   Google Scholar

[13]

R. Osserman, Global properties of minimal surfaces in $E^3$ and $E^n$,, \emph{Ann. of Math. (2)}, 80 (1964), 340.  doi: 10.2307/1970396.  Google Scholar

[14]

R. Schoen, Uniqueness, symmetry, and embeddedness of minimal surfaces,, \emph{J. Differential Geom.}, 18 (1983), 791.   Google Scholar

[15]

A. Tarski, A Decision Method for an Elementary Algebra and Geometry,, 2nd edition, (1951).   Google Scholar

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