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ISSN 1088-6826(e) ISSN 0002-9939(p)

Weak continuity of the Gauss-Codazzi-Ricci system for isometric embedding

Author(s): Gui-Qiang Chen; Marshall Slemrod; Dehua Wang
Journal: Proc. Amer. Math. Soc. 138 (2010), 1843-1852.
MSC (2010): Primary 53C42, 53C21, 53C45, 35L65, 35M10, 35B35; Secondary 53C24, 57R40, 57R42, 58J32
Posted: December 31, 2009
MathSciNet review: 2587469
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Abstract: We establish the weak continuity of the Gauss-Coddazi-Ricci system for isometric embedding with respect to the uniform -bounded solution sequence for , which implies that the weak limit of the isometric embeddings of the manifold in a fixed coordinate chart is an isometric immersion. More generally, we establish a compensated compactness framework for the Gauss-Codazzi-Ricci system in differential geometry. That is, given any sequence of approximate solutions to this system which is uniformly bounded in and has reasonable bounds on the errors made in the approximation (the errors are confined in a compact subset of $H^{-1}_{\text{loc}}$ ), the approximating sequence has a weakly convergent subsequence whose limit is a solution of the Gauss-Codazzi-Ricci system. Furthermore, a minimizing problem is proposed as a selection criterion. For these, no restriction on the Riemann curvature tensor is made.

References:

1.: J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1976/77), 337-403. MR 0475169 (57:14788)
2.: E. Berger, R. Bryant, and P. Griffiths, Some isometric embedding and rigidity results for Riemannian manifolds, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), 4657-4660. MR 627257 (82h:53074)
3.: E. Berger, R. Bryant, and P. Griffiths, The Gauss equations and rigidity of isometric embeddings, Duke Math. J. 50 (1983), 803-892. MR 714831 (85k:53056)
4.: R. L. Bryant, P. A. Griffiths, and D. Yang, Characteristics and existence of isometric embeddings, Duke Math. J. 50 (1983), 893-994. MR 726313 (85d:53027)
5.: Yu. D. Burago and S. Z. Shefel, The geometry of surfaces in Euclidean spaces, Geometry III, 1-85, Encyclopaedia Math. Sci., 48, Burago and Zalggaller (eds.), Springer-Verlag: Berlin, 1992. MR 1306734
6.: E. Cartan, Sur la possibilité de plonger un espace Riemannian donné dans un espace Euclidien, Ann. Soc. Pol. Math. 6 (1927), 1-7.
7.: B.-Y. Chen, Classification of locally symmetric spaces which admit a totally umbilical hypersurface, Soochow J. Math. 6 (1980), 39-48. MR 626317 (82j:53096)
8.: G.-Q. Chen, M. Slemrod, and D. Wang, Isometric immersions and compensated compactness, Commun. Math. Phys., in press.
9.: B. Dacorogna, Weak Continuity and Weak Lower Semicontinuity of Nonlinear Functionals, Springer-Verlag: Berlin, 1982. MR 658130 (84f:49020)
10.: C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, second edition, Springer-Verlag: Berlin, 2005. MR 2169977 (2006d:35159)
11.: B. A. DiDonna, T. A. Witten, S. C. Venkataramani, and E. M. Kramer, Singularities, structures, and scaling in deformed m-dimensional elastic manifolds. Phys. Rev. E (3) 65 (2002), 016603, 1-25. MR 1877621 (2002k:82091)
12.: L. P. Eisenhart, Riemannian Geometry, eighth printing, Princeton University Press: Princeton, NJ, 1997. MR 1487892 (98h:53001)
13.: L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS-RCSM, 74, Amer. Math. Soc.: Providence, RI, 1990. MR 1034481 (91a:35009)
14.: H. F. Goenner, On the interdependency of the Gauss-Codazzi-Ricci equations of local isometric embedding, General Relativity and Gravitation 8 (1977), 139-145. MR 0467606 (57:7462)
15.: R. Greene, Isometric embeddings of Riemannian and pseudo-Riemannian manifolds, Memoirs Amer. Math. Soc., 97, Amer. Math. Soc.: Providence, RI, 1970. MR 0262980 (41:7585)
16.: M. Günther, On the perturbation problem associated to isometric embeddings of Riemannian manifolds, Ann. Global Anal. Geom. 7 (1989), 69-77. MR 1029846 (91a:58023)
17.: Q. Han and J.-X. Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Mathematical Surveys and Monographs, 130, Amer. Math. Soc.: Providence, RI, 2006. MR 2261749 (2008e:53055)
18.: M. Janet, Sur la possibilité de plonger un espace Riemannian donné dans un espace Euclidien, Ann. Soc. Pol. Math. 5 (1926), 38-43.
19.: C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer: Berlin, New York, 1966. MR 0202511 (34:2380)
20.: S. Müller, A surprising higher integrability property of mappings with positive determinant, Bull. Amer. Math. Soc. (N.S.) 21 (1989), 245-248. MR 999618 (90c:49022)
21.: F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), 489-507. MR 506997 (80h:46043a)
22.: F. Murat, Compacité par compensation. II, in: Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), pp. 245-256, Pitagora: Bologna, 1979. MR 533170 (80h:46043b)
23.: G. Nakamura and Y. Maeda, Local isometric embedding problem of Riemannian -manifold into $\mathbb{R}\sp 6$ , Proc. Japan Acad. Ser. A: Math. Sci. 62 (1986), 257-259. MR 868813 (88b:53026)
24.: J. Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. (2) 63 (1956), 20-63. MR 0075639 (17:782b)
25.: M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish, Inc.: Boston, MA, Vols. I-II, 1970; Vols. III-V, 1975. MR 0532830 (82g:53003a), MR 0532831 (82g:53003b); MR 0532832 (82g:53003c), MR 0532833 (82g:53003d), MR 0532834 (82g:53003e)
26.: L. Tartar, Compensated compactness and applications to partial differential equations, in: Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, pp. 136-212, Res. Notes in Math., 39, Pitman: Boston, MA; London, 1979. MR 584398 (81m:35014)
27.: L. Tartar, The compensated compactness method applied to systems of conservation laws, in: Systems of Nonlinear Partial Differential Equations (Oxford, 1982), 263-285, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 111, Reidel: Dordrecht, 1983. MR 725524 (85e:35079)
28.: T. A. Witten, Stress focusing in elastic sheets, Reviews of Modern Physics 79 (2007), 643-675. MR 2326936 (2008e:74052)

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Additional Information:

Gui-Qiang Chen
Affiliation: School of Mathematical Sciences, Fudan University, Shanghai 200433, People's Republic of China - and - Department of Mathematics, Northwestern University, Evanston, Illinois 60208 - and - Mathematical Institute, University of Oxford, Oxford, OX1 3LB, United Kingdom
Email: gqchen@math.northwestern.edu

Marshall Slemrod
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706 - and - Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260 - and - Department of Mathematical Sciences, Korean Advanced Institute for Science and Technology, Daejeon, Republic of Korea
Email: slemrod@math.wisc.edu

Dehua Wang
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: dwang@math.pitt.edu

DOI: 10.1090/S0002-9939-09-10187-9
PII: S 0002-9939(09)10187-9
Keywords: Weak continuity, Gauss-Codazzi-Ricci system, isometric embedding, weak convergence, approximate solutions, compensated compactness, Div-Curl lemma, minimization problem, selection criterion, Riemann curvature tensor
Received by editor(s): August 14, 2009,
Received by editor(s) in revised form: August 27, 2009
Posted: December 31, 2009
Communicated by: Walter Craig
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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