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, 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.

[1]

V. Afraimovich, T.R. Young. Multipliers of homoclinic orbits on surfaces and characteristics of associated invariant sets. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 691-704. doi: 10.3934/dcds.2000.6.691

[2]

Xinru Ji, Bingjie Wang, Jigao Yan, Dongya Cheng. Asymptotic estimates for finite-time ruin probabilities in a generalized dependent bidimensional risk model with CMC simulations. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022036

[3]

Sishu Shankar Muni, Robert I. McLachlan, David J. W. Simpson. Unfolding globally resonant homoclinic tangencies. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 4013-4030. doi: 10.3934/dcds.2022043

[4]

Danijela Damjanovic, James Tanis, Zhenqi Jenny Wang. On globally hypoelliptic abelian actions and their existence on homogeneous spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6747-6766. doi: 10.3934/dcds.2020164

[5]

Ambros M. Gleixner, Harald Held, Wei Huang, Stefan Vigerske. Towards globally optimal operation of water supply networks. Numerical Algebra, Control and Optimization, 2012, 2 (4) : 695-711. doi: 10.3934/naco.2012.2.695

[6]

Fanni M. Sélley. Symmetry breaking in a globally coupled map of four sites. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3707-3734. doi: 10.3934/dcds.2018161

[7]

G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini. Globally stable quasistatic evolution in plasticity with softening. Networks and Heterogeneous Media, 2008, 3 (3) : 567-614. doi: 10.3934/nhm.2008.3.567

[8]

Weijun Zhou. A globally convergent BFGS method for symmetric nonlinear equations. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1295-1303. doi: 10.3934/jimo.2021020

[9]

Alfonso Artigue. Expansive flows of surfaces. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 505-525. doi: 10.3934/dcds.2013.33.505

[10]

Kariane Calta, John Smillie. Algebraically periodic translation surfaces. Journal of Modern Dynamics, 2008, 2 (2) : 209-248. doi: 10.3934/jmd.2008.2.209

[11]

Anton Petrunin. Correction to: Metric minimizing surfaces. Electronic Research Announcements, 2018, 25: 96-96. doi: 10.3934/era.2018.25.010

[12]

Anton Petrunin. Metric minimizing surfaces. Electronic Research Announcements, 1999, 5: 47-54.

[13]

Siran Li, Jiahong Wu, Kun Zhao. On the degenerate boussinesq equations on surfaces. Journal of Geometric Mechanics, 2020, 12 (1) : 107-140. doi: 10.3934/jgm.2020006

[14]

Yong Lin, Gábor Lippner, Dan Mangoubi, Shing-Tung Yau. Nodal geometry of graphs on surfaces. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1291-1298. doi: 10.3934/dcds.2010.28.1291

[15]

Gabriele Beltramo, Primoz Skraba, Rayna Andreeva, Rik Sarkar, Ylenia Giarratano, Miguel O. Bernabeu. Euler characteristic surfaces. Foundations of Data Science, 2021  doi: 10.3934/fods.2021027

[16]

Eduard Duryev, Charles Fougeron, Selim Ghazouani. Dilation surfaces and their Veech groups. Journal of Modern Dynamics, 2019, 14: 121-151. doi: 10.3934/jmd.2019005

[17]

Anh N. Le. Sublacunary sets and interpolation sets for nilsequences. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1855-1871. doi: 10.3934/dcds.2021175

[18]

Alexandre Girouard, Iosif Polterovich. Upper bounds for Steklov eigenvalues on surfaces. Electronic Research Announcements, 2012, 19: 77-85. doi: 10.3934/era.2012.19.77

[19]

Seung Won Kim, P. Christopher Staecker. Dynamics of random selfmaps of surfaces with boundary. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 599-611. doi: 10.3934/dcds.2014.34.599

[20]

Roberto Paroni, Podio-Guidugli Paolo, Brian Seguin. On the nonlocal curvatures of surfaces with or without boundary. Communications on Pure and Applied Analysis, 2018, 17 (2) : 709-727. doi: 10.3934/cpaa.2018037

2020 Impact Factor: 0.929

Metrics

  • PDF downloads (131)
  • HTML views (0)
  • Cited by (1)

[Back to Top]