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Ends of negatively curved surfaces in Euclidean space

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Abstract

We examine the geometry of a complete, negatively curved surface isometrically embedded in \({\mathbb{R}^3}\). We are especially interested in the behavior of the ends of the surface and its limit set at infinity. Various constructions are developed, and a classification theorem is obtained, showing that every possible end type for a topologically finite surface with at least one bowl end arises, as well as all infinite type surfaces with a single nonannular end. Some other examples are given with oddly behaved bowl ends.

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References

  1. Brandt I.S.: Certain properties of surfaces with slowly changing negative outer curvature in a Riemannian space. Mat. Sb. (N.S.) 83(125), 313–324 (1970)

    MathSciNet  Google Scholar 

  2. Burago Y.D.: Realization of a metrized two-dimensional manifold by a surface in E 3. Sov. Math. Dokl. 1, 1364–1365 (1960)

    MathSciNet  Google Scholar 

  3. Burago, Y.D.: Existence on a noncompact surface of a complete metric with given curvature. Ukrain. Geom. Sb. 25, 8–11, 140 (1982)

    Google Scholar 

  4. Burago D.Y.: Unboundedness in Euclidean space of a horn with a finite positive part of the curvature. Mat. Zametki 36(2), 229–237 (1984)

    MathSciNet  Google Scholar 

  5. Burago, Y.D., Zalgaller, V.A. (eds.): Geometry. III, theory of surfaces, Encyclopaedia of Mathematical Sciences, vol. 48, Springer-Verlag, Berlin (1992). (A translation of Current problems in mathematics. Fundamental directions. vol. 48 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow (1989) Translation by E. Primrose)

  6. Bakel′man, I.Ya., Verner, A.L., Kantor, B.E.: Vvedenie v differentsialnuyu geometriyu “v tselom”. Izdat. “Nauka”, Moscow (1973)

  7. Connell C., Ghomi M.: Topology of negatively curved real affine algebraic surfaces. J. Reine Angew. Math. 624, 1–26 (2008)

    MATH  MathSciNet  Google Scholar 

  8. Efimov, N.V.: Surfaces with slowly varying negative curvature. Uspekhi Mat. Nauk 21(5, suppl. 131), 3–58 (1966)

    Google Scholar 

  9. Ghomi, M.: A negatively curved genus one surface with one end. Personal Communication (2007)

  10. Kantor, B.E.: Reconstruction of smooth surfaces from given curvature functions. Questions of differential geometry “in the large”, pp. 29–46. Leningrad. Gos. Ped. Inst., Leningrad (1983)

  11. Kon-Fossen, S.È.: Nekotorye voprosy differentsialnoi geometrii v tselom. In: Efimov, N.V. (ed.) Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow (1959)

  12. Martinez-Maure Y.: Contre-exemple à une caractérisation conjecturée de la sphère. C. R. Acad. Sci. Paris Sér. I Math. 332(1), 41–44 (2001)

    MATH  MathSciNet  Google Scholar 

  13. Milnor T.K.: Efimov’s theorem about complete immersed surfaces of negative curvature. Adv. Math. 8, 474–543 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  14. Rozendorn, È.R.: Weakly irregular surfaces of negative curvature. Uspekhi Mat. Nauk 21(5, suppl. 131), 59–116 (1966)

    Google Scholar 

  15. Rozendorn, È.R.: Bounded complete weakly nonregular surfaces with negative curvature bounded away from zero. Mat. Sb. (N.S.) 116(158, suppl. 4), 558–567, 607 (1981)

  16. Verner A.L.: The extrinsic geometry of the simplest complete surfaces of non-positive curvature. I. Mat. Sb. (N.S.) 74(116), 218–240 (1967)

    MathSciNet  Google Scholar 

  17. Verner A.L.: Concerning S. Cohn-Vossen’s theorem on the integral curvature of complete surfaces. Sibirsk. Mat. Ž. 9, 199–203 (1968)

    MATH  MathSciNet  Google Scholar 

  18. Verner A.L.: The extrinsic geometry of the simplest complete surfaces of non-positive curvature. II. Mat. Sb. (N.S.) 75(117), 112–139 (1968)

    MathSciNet  Google Scholar 

  19. Verner A.L.: Constricting saddle surfaces. Sibirsk. Mat. Ž. 11, 750–769 (1970)

    MATH  MathSciNet  Google Scholar 

  20. Verner A.L.: Unboundedness of a hyperbolic horn in Euclidean space. Sibirsk. Mat. Ž. 11, 20–29 (1970)

    MathSciNet  Google Scholar 

  21. Vinogradskiĭ A.S.: Boundary properties of surfaces with slowly changing negative curvature. Mat. Sb. (N.S.) 82(124), 285–299 (1970)

    MathSciNet  Google Scholar 

  22. Vinogradskiĭ A.S.: Causes for the nonextendability of surfaces of negative curvature bounded away from zero beyond a smooth boundary. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 25(5), 83–86 (1970)

    Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, Indiana University, 831 East 3rd St., Bloomington, IN, 47405, USA

    Chris Connell

  2. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA, 02139-4307, USA

    John Ullman

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  1. Chris Connell
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  2. John Ullman
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Correspondence to Chris Connell.

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Connell, C., Ullman, J. Ends of negatively curved surfaces in Euclidean space. manuscripta math. 131, 275–303 (2010). https://doi.org/10.1007/s00229-009-0324-x

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  • DOI: https://doi.org/10.1007/s00229-009-0324-x

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