Skip to main content

Isometric Immersions of Surfaces with Two Classes of Metrics and Negative Gauss Curvature

Abstract

The isometric immersion of two-dimensional Riemannian manifolds or surfaces with negative Gauss curvature into the three-dimensional Euclidean space is studied in this paper. The global weak solutions to the Gauss-Codazzi equations with large data in \({L^{\infty}}\) are obtained through the vanishing viscosity method and the compensated compactness framework. The \({L^{\infty}}\) uniform estimate and H −1 compactness are established through a transformation of state variables and construction of proper invariant regions for two types of given metrics including the catenoid type and the helicoid type. The global weak solutions in \({L^{\infty}}\) to the Gauss-Codazzi equations yield the C 1,1 isometric immersions of surfaces with the given metrics.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Ball, J.M.: A Version of the Fundamental Theorem for Young Measures. Lect. Notes Phys. 344, 207–215, Springer, Berlin (1989)

  2. 2.

    Berger E., Bryant R., Griffiths P.: Some isometric embedding and rigidity results for Riemannian manifolds. Proc. Nat. Acad. Sci. 78, 4657–4660 (1981)

    MATH  MathSciNet  Article  ADS  Google Scholar 

  3. 3.

    Berger E., Bryant R., Griffiths P.: Characteristics and rigidity of isometric embeddings. Duke Math. J. 50, 803–892 (1983)

    MATH  MathSciNet  Article  Google Scholar 

  4. 4.

    Bryant R.L., Griffiths P.A., Yang D.: Characteristics and existence of isometric embeddings. Duke Math. J. 50, 893–994 (1983)

    MATH  MathSciNet  Article  Google Scholar 

  5. 5.

    Burago, Y.D., Shefel, S.Z.: The geometry of surfaces in Euclidean spaces. Geometry III, 1–85, Encyclopaedia Math. Sci., 48, (Eds. Burago and Zalggaller), Springer, Berlin, 1992

  6. 6.

    Cartan, E.: Sur la possibilité de plonger un espace Riemannian donné dans un espace Euclidien. Ann. Soc. Pol. Math. 6, 1–7 (1927)

  7. 7.

    Chen, G.-Q.: Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (III). Acta Math. Sci. 6, 75–120, 1986 (in English); 8, 243–276 (1988) (in Chinese)

  8. 8.

    Chen G.-Q., Slemrod M., Wang D.: Isomeric immersion and compensated compactness. Commun. Math. Phys. 294, 411–437 (2010)

    MATH  MathSciNet  Article  ADS  Google Scholar 

  9. 9.

    Chen G.-Q., Slemrod M., Wang D.: Weak continuity of the Gauss-Codazzi-Ricci system for isometric embedding. Proc. Amer. Math. Soc. 138, 1843–1852 (2009)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Chen, G.-Q., Slemrod, M., Wang, D.: Entropy, elasticity, and the isometric embedding problem: \({{M^{3} \to R^{6}}}\) . In: Hyperbolic Conservation Laws and Related Analysis with Applications, Springer Proceedings in Mathematics and Statistics 49, 95–112, Springer, Berlin, 2014

  11. 11.

    Christoforou C.: BV weak solutions to Gauss-Codazzi system for isometric immersions. J. Differ. Equs. 252(3), 2845–2863 (2012)

    MATH  MathSciNet  Article  ADS  Google Scholar 

  12. 12.

    Codazzi, D.: Sulle coordinate curvilinee d’una superficie dello spazio. Ann. Math. Pura Appl. 2, 101–119 (1860)

  13. 13.

    Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, 2nd edition, Springer, Berlin, 2005

  14. 14.

    Ding, X., Chen, G.-Q., Luo, P.: Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics (I)–(II). Acta Math. Sci. 5, 483–500 (1985) 501–540 (in English); 7, 467–480 (1987) 8, 61–94 (1988) (in Chinese)

  15. 15.

    DiPerna, R.J.: Convergence of viscosity method for isentropic gas dynamics. Commun. Math. Phys. 91, 1–30 (1983)

  16. 16.

    DiPerna, R.J.: Compensated compactness and general systems of conservation laws. Trans. Am. Math. Soc. 292, 383–420 (1985)

  17. 17.

    do Carmo, M.P.: Riemannian Geometry. Transl. by F. Flaherty, Birkhäuser: Boston MA (1992)

  18. 18.

    Dong, G.-C.: The semi-global isometric imbedding in R 3 of two-dimensional Riemannian manifolds with Gaussian curvature changing sign cleanly. J. Partial Diff. Eqs. 6, 62–79 (1993)

  19. 19.

    Efimov, N.V.: The impossibility in Euclideam 3-space of a complete regular surface with a negative upper bound of the Gaussian curvature. Dokl. Akad. Nauk SSSR (N.S.) 150, 1206–1209 (1963) Soviet Math. Dokl. 4, 843–846 (1963)

  20. 20.

    Efimov, N.V.: Surfaces with slowly varying negative curvature. Russ. Math. Sur. 21, 1–55 (1966)

  21. 21.

    Ghrist, R.: Configuration spaces, braids, and robotics. Lecture Note Series. Inst. Math. Sci., NUS, vol. 19, World Scientific, 263–304

  22. 22.

    Eisenhart, L.P.: Riemannian Geometry. Eighth Printing. Princeton University Press, Princeton, 1997

  23. 23.

    Evans, L.C.: Weak Convergence Methods for Nonlinear Partial Differential Equations. CBMS-RCSM, 74. AMS: Providence, RI, 1990

  24. 24.

    Evans, L.C.: Partial Differential Equations. Providence, RI, Amer. Math. Soc., 1998

  25. 25.

    Gromov, M.: Partial Differential Relations. Springer, Berlin, 1986

  26. 26.

    Han, Q.: On isometric embedding of surfaces with Gauss curvature changing sign cleanly. Commun. Pure Appl. Math. 58, 285–295 (2005)

  27. 27.

    Han, Q., Hong, J.-X.: Isomeric embedding of Riemannian manifolds in Euclidean spaces. Providence, RI, Amer. Math. Soc., 2006

  28. 28.

    Hong, J.-X.: Realization in \({\mathbb{R}^3}\) of complete Riemannian manifolds with negative curvature. Commun. Anal. Geom., 1, 487–514 (1993)

  29. 29.

    Huang, F., Wang, Z.: Convergence of viscosity for isothermal gas dynamics. SIAM J. Math. Anal. 34, 595–610 (2002)

  30. 30.

    Janet, M.: Sur la possibilité de plonger un espace Riemannian donné dans un espace Euclidien. Ann. Soc. Pol. Math. 5, 38–43 (1926)

  31. 31.

    Lin, C.-S.: The local isometric embedding in \({\mathbb{R}^3}\) of 2-dimensional Riemannian manifolds with Gaussian curvature changing sign cleanly. Commun. Pure Appl. Math. 39, 867–887 (1986)

  32. 32.

    Lions, P.-L., Perthame, B., Souganidis, P.: Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Commun. Pure Appl. Math. 49, 599–638 (1996)

  33. 33.

    Lions, P.-L., Perthame, B., Tadmor, E.: Kinetic formulation of the isentropic gas dynamics and p-systems. Commun. Math. Phys. 163, 169–172 (1994)

  34. 34.

    Mainardi, G.: Su la teoria generale delle superficie. Giornale dell’ Istituto Lombardo 9, 385–404 (1856)

  35. 35.

    Mardare, S.: The foundamental theorem of theorey for surfaces with little regularity. J. Elast. 73, 251–290 (2003)

  36. 36.

    Mardare, S.: On Pfaff systems with L p coefficients and their applications in differential geometry. J. Math. Pure Appl. 84, 1659–1692 (2005)

  37. 37.

    Murat, F.: Compacite par compensation. Ann. Suola Norm. Pisa (4), 5, 489–507 (1978)

  38. 38.

    Nakamura, G., Maeda, Y.: Local isometric embedding problem of Riemannian 3-manifold into R 6. Proc. Jpn. Acad. Ser. A Math. Sci. 62(7), 257–259 (1986)

  39. 39.

    Nakamura, G., Maeda, Y.: Local smooth isometric embeddings of low-dimensional Riemannian manifolds into Euclidean spaces. Trans. Amer. Math. Soc. 313(1), 1–51 (1989)

  40. 40.

    Nash, J.: The imbedding problem for Riemannian manifolds. Ann. Math. (2), 63, 20–63

  41. 41.

    Peterson, K.M.: Ueber die Biegung der Flächen. Dorpat. Kandidatenschrift, 1853

  42. 42.

    Polyanin, A.D., Zaitsev, V.F.: Handbook of Exact Solutions for Ordinary Differential Equations. Chapman & Hall/CRC, Bergman, 2002

  43. 43.

    Poole, T.E.: The local isometric embedding problem for 3-dimensional Riemannian manifolds with cleanly vanishing curvature. Comm. Partial Differ. Equs. 35, 1802–1826 (2010)

  44. 44.

    Poznyak, È.G., Shikin, E.V.: Small parameters in the theory of isometric imbeddings of two-dimensional Riemannian manifolds in Euclidean spaces. In: Some Questions of Differential Geometry in the Large. 2, 176, 151–192, Amer. Math. Soc. Transl. Ser. AMS: Providence, RI, 1996

  45. 45.

    Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Springer, Berlin, 1984

  46. 46.

    Rozendorn, È.R.: Surfaces of negative curvature. In: Geometry, III, 87–178, 251–256, Encyclopaedia Math. Sci. 48, Springer, Berlin, 1992

  47. 47.

    Smoller, J.: Shock Waves and Reaction-Diffusion Equations. Springer, New York, 1994

  48. 48.

    Tartar, L.: Compensated compactness and applications to partial differential equations. In: Nonlinear Analysis and Mechanics, Heriot-Watt Symposium IV, Res. Notes in Math. 39, pp. 136–212 Pitman, Boston-London, 1979

  49. 49.

    Vaziri, A., Mahedevan, L.: Localized and extended deformations of elastic shells. Proc. Natl. Acad. Sci, USA 105, 7913–7918 (2008)

  50. 50.

    Yau, S.-T.: Review of geometry and analysis. In: Mathematics: Frontiers and Perspectives, pp. 353–401, International Mathematics Union, (Eds. V. Arnold, M. Atiyah, P. Lax and B. Mazur) AMS: Providence, 2000

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Feimin Huang.

Additional information

Communicated by C. Dafermos

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cao, W., Huang, F. & Wang, D. Isometric Immersions of Surfaces with Two Classes of Metrics and Negative Gauss Curvature. Arch Rational Mech Anal 218, 1431–1457 (2015). https://doi.org/10.1007/s00205-015-0885-7

Download citation

Keywords

  • Weak Solution
  • Riemannian Manifold
  • Invariant Region
  • Isometric Immersion
  • Isometric Embedding