January  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

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

Yannan Liu, Linfen Cao. Lifespan theorem and gap lemma for the globally constrained Willmore flow. Communications on Pure & Applied Analysis, 2014, 13 (2) : 715-728. doi: 10.3934/cpaa.2014.13.715

[12]

Xian-Gao Liu, Jie Qing. Globally weak solutions to the flow of compressible liquid crystals system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 757-788. doi: 10.3934/dcds.2013.33.757

[13]

Tomás Caraballo, Peter E. Kloeden, José Real. Invariant measures and Statistical solutions of the globally modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 761-781. doi: 10.3934/dcdsb.2008.10.761

[14]

Hayato Chiba. Continuous limit and the moments system for the globally coupled phase oscillators. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1891-1903. doi: 10.3934/dcds.2013.33.1891

[15]

Zhichuan Zhu, Bo Yu, Li Yang. Globally convergent homotopy method for designing piecewise linear deterministic contractual function. Journal of Industrial & Management Optimization, 2014, 10 (3) : 717-741. doi: 10.3934/jimo.2014.10.717

[16]

Guolin Yu. Topological properties of Henig globally efficient solutions of set-valued problems. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 309-316. doi: 10.3934/naco.2014.4.309

[17]

Nuno Costa Dias, Andrea Posilicano, João Nuno Prata. Self-adjoint, globally defined Hamiltonian operators for systems with boundaries. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1687-1706. doi: 10.3934/cpaa.2011.10.1687

[18]

Pedro Marín-Rubio, Antonio M. Márquez-Durán, José Real. Pullback attractors for globally modified Navier-Stokes equations with infinite delays. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 779-796. doi: 10.3934/dcds.2011.31.779

[19]

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

[20]

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

2018 Impact Factor: 0.263

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (0)

[Back to Top]