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

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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.

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References

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

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  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

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Authors and Affiliations

  1. Institute of Applied Mathematics, AMSS, CAS, Beijing, 100190, China

    Wentao Cao & Feimin Huang

  2. College of Mathematics and Computer Science, Hunan Normal University, Changsha, 410081, China

    Feimin Huang

  3. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Dehua Wang

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  1. Wentao Cao
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  2. Feimin Huang
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  3. Dehua Wang
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Correspondence to Feimin Huang.

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Communicated by C. Dafermos

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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

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