Connecting topological Hopf singularities

Robert Hardt; Tristan Rivière

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2003)

  • Volume: 2, Issue: 2, page 287-344
  • ISSN: 0391-173X

Abstract

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Smooth maps between riemannian manifolds are often not strongly dense in Sobolev classes of finite energy maps, and an energy drop in a limiting sequence of smooth maps often is accompanied by the production (or bubbling) of an associated rectifiable current. For finite 2-energy maps from the 3 ball to the 2 sphere, this phenomenon has been well-studied in works of Bethuel-Brezis-Coron and Giaquinta-Modica-Soucek where a finite mass 1 dimensional rectifiable current occurs whose boundary is the algebraic singular set of the limiting map. The relevant algebraic object here is π 2 ( S 2 ) which provides both the obstruction to strong approximation by smooth maps and the topological structure to the bubbling set and the singular set. With higher homotopy groups, new phenomena occur. For π 3 ( S 2 ) and the related case of finite 3-energy maps from the 4 Ball to the 2 sphere, there are examples with bubbled objects that no longer have finite mass. We define a new object, a scan, which generalizes a current but still occurs naturally in bubbling while automatically providing the topological connection between the singularities of the limit map. The bubbled scans, which are found via a new compactness theorem, again enjoy a representation using a finite measure 1 rectifiable set and an integer density function which is now however only L 3 / 4 (rather than L 1 ) integrable.

How to cite

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Hardt, Robert, and Rivière, Tristan. "Connecting topological Hopf singularities." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 2.2 (2003): 287-344. <http://eudml.org/doc/84503>.

@article{Hardt2003,
abstract = {Smooth maps between riemannian manifolds are often not strongly dense in Sobolev classes of finite energy maps, and an energy drop in a limiting sequence of smooth maps often is accompanied by the production (or bubbling) of an associated rectifiable current. For finite 2-energy maps from the 3 ball to the 2 sphere, this phenomenon has been well-studied in works of Bethuel-Brezis-Coron and Giaquinta-Modica-Soucek where a finite mass 1 dimensional rectifiable current occurs whose boundary is the algebraic singular set of the limiting map. The relevant algebraic object here is $\pi _2(S^2)$ which provides both the obstruction to strong approximation by smooth maps and the topological structure to the bubbling set and the singular set. With higher homotopy groups, new phenomena occur. For $\pi _3(S^2)$ and the related case of finite 3-energy maps from the 4 Ball to the 2 sphere, there are examples with bubbled objects that no longer have finite mass. We define a new object, a scan, which generalizes a current but still occurs naturally in bubbling while automatically providing the topological connection between the singularities of the limit map. The bubbled scans, which are found via a new compactness theorem, again enjoy a representation using a finite measure $1$ rectifiable set and an integer density function which is now however only $L^\{3/4\}$ (rather than $L^1$) integrable.},
author = {Hardt, Robert, Rivière, Tristan},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {2},
pages = {287-344},
publisher = {Scuola normale superiore},
title = {Connecting topological Hopf singularities},
url = {http://eudml.org/doc/84503},
volume = {2},
year = {2003},
}

TY - JOUR
AU - Hardt, Robert
AU - Rivière, Tristan
TI - Connecting topological Hopf singularities
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2003
PB - Scuola normale superiore
VL - 2
IS - 2
SP - 287
EP - 344
AB - Smooth maps between riemannian manifolds are often not strongly dense in Sobolev classes of finite energy maps, and an energy drop in a limiting sequence of smooth maps often is accompanied by the production (or bubbling) of an associated rectifiable current. For finite 2-energy maps from the 3 ball to the 2 sphere, this phenomenon has been well-studied in works of Bethuel-Brezis-Coron and Giaquinta-Modica-Soucek where a finite mass 1 dimensional rectifiable current occurs whose boundary is the algebraic singular set of the limiting map. The relevant algebraic object here is $\pi _2(S^2)$ which provides both the obstruction to strong approximation by smooth maps and the topological structure to the bubbling set and the singular set. With higher homotopy groups, new phenomena occur. For $\pi _3(S^2)$ and the related case of finite 3-energy maps from the 4 Ball to the 2 sphere, there are examples with bubbled objects that no longer have finite mass. We define a new object, a scan, which generalizes a current but still occurs naturally in bubbling while automatically providing the topological connection between the singularities of the limit map. The bubbled scans, which are found via a new compactness theorem, again enjoy a representation using a finite measure $1$ rectifiable set and an integer density function which is now however only $L^{3/4}$ (rather than $L^1$) integrable.
LA - eng
UR - http://eudml.org/doc/84503
ER -

References

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  1. [AK] L. Ambrosio – B. Kirchheim, Currents in metric spaces, Acta Math. 185 (2000), 1-80. Zbl0984.49025MR1794185
  2. [Be1] F. Bethuel, The approximation problem for Sobolev maps between two manifolds, Acta Math. 167 (1991), 153-206. Zbl0756.46017MR1120602
  3. [Be2] F. Bethuel, A characterization of maps in H 1 ( B 3 , S 2 ) which can be approximated by smooth maps, Ann. Inst. H. Poincaré Anal. Nonlinaire 7 (1990), 269-286. Zbl0708.58004MR1067776
  4. [Be3] F. Bethuel, Approximations in trace spaces defined between manifolds, Nonlinear Anal. 24 (1995), 121-130. Zbl0824.58011MR1308474
  5. [BBC] F. Bethuel – H. Brezis – J.-M. Coron, Relaxed energies for harmonic maps, In: “Variational methods", 3–52, Progr. Nonlin. Diff. Eqns. Appl. 4, Birkhäuser Boston, 1990. Zbl0793.58011MR1205144
  6. [BCL] H. Brezis – J.-M. Coron, – E. Lieb, Harmonic maps with defects, Comm. Math. Phys. 107, no. 4 (1986), 649-705. Zbl0608.58016MR868739
  7. [BL] H. Brezis – Y.Y. Li, Topology and Sobolev spaces, preprint, 2000. Zbl1001.46019MR1784915
  8. [BZ] F. Bethuel – X.M. Zheng, Density of smooth functions between two manifolds in Sobolev spaces, J. Funct. Anal. 80, no. 1 (1988), 60-75. Zbl0657.46027MR960223
  9. [F] H. Federer, "Geometric measure theory", Springer-Verlag, Berlin and New York, 1969. Zbl0176.00801MR257325
  10. [Fl] Fleming, flat chains over a coefficient group, Trans. Amer. Math. Soc. 121 (1966), 160-186. Zbl0136.03602MR185084
  11. [G] M. Giaquinta, “Multiple integrals in the calculus of variations and elliptic systems”, Princeton University Press, Princeton, 1983. Zbl0516.49003MR717034
  12. [GMS1] M. Giaquinta – G. Modica – J. Souček, The Dirichlet energy of mappings with values into the sphere, Manuscripta Math. 65, no. 4 (1989), 489-507. Zbl0678.49006MR1019705
  13. [GMS2] M. Giaquinta – G. Modica – J. Souček, "Cartesian currents in the calculus of variations". I, II, Springer-Verlag, Berlin, 1998. Zbl0914.49001MR1645086
  14. [Gr] M. Gromov, "Metric structures for Riemannian and non-Riemannian spaces", Progress in mathematics 152, Birkhaüser, Boston, 1999. Zbl0953.53002MR1699320
  15. [H] R. Hardt, Uniqueness of nonparametric area minimizing currents, Indiana Univ. Math. J. 26 (1977), 65-71. Zbl0333.49043MR451154
  16. [HD] R. Hardt – T. DePauw, Size minimization and approximating problems, to appear in Calc. Var. Partial Differential Equations. Zbl1022.49026MR1993962
  17. [He] F. Hélein, "Harmonic maps, conservation laws, and moving frames", Diderot Press, Paris, 1997. Zbl1125.58300MR1913803
  18. [HgL1] F. Hang – Lin, Topology of Sobolev mappings, Math. Res. Lett. 8 (2001), 321-330. Zbl1049.46018MR1839481
  19. [HgL2] F. Hang – F. Lin, Topology of Sobolev mappings II, preprint, 2001. Zbl1061.46032MR1839481
  20. [HL1] R. Hardt – F.H. Lin, A remark on H 1 mappings, Manuscripta Math. 56 (1986), 1-10. Zbl0618.58015MR846982
  21. [HL2] R. Hardt – F.H. Lin, Mappings minimizing the L p norm of the gradient, Comm. Pure. Appl. Math. 15 (1987), 555-588. Zbl0646.49007MR896767
  22. [HR1] R. Hardt – T. Rivière, Ensembles singuliers topologiques dans les espaces fonctionnels entre variétés, Exp. VII, Sémin. Équ. Dériv. Part., École Polytech., Palaiseau, 2001. Zbl1057.58003MR1860679
  23. [HR2] R. Hardt – T. Rivière, Bubbling phenomena for maps in W 2 , 2 ( 𝔹 5 , § 3 ) , in preparation. 
  24. [HR3] R. Hardt – T. Rivière, Connecting singularities of arbitrary rational homotopy type, in preparation. Zbl1148.58003
  25. [I1] T. Isobe, Characterization of the strong closure of C ( B 4 , S 2 ) in W 1 , p ( B 4 , S 2 ) ( 16 / 5 p l t ; 4 ) , J. Math. Anal. Appl. 190 (1995), 361-372. Zbl0838.46028MR1318399
  26. [I2] T. Isobe, Some new properties of Sobolev mappings: intersection theoretical approach, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), 337-358. Zbl0899.46021MR1447955
  27. [M] D. Mucci, A characterization of graphs which can be approximated in area by smooth graphs, To appear in the J. Eur. Math. Soc. Zbl0996.49025MR1812123
  28. [PR] M. R. Pakzad – T. Rivière, Weak density of smooth maps for the Dirichlet energy between manifolds, preprint, 2000. Zbl1028.58008MR1978496
  29. [R1] T. Rivière, Minimizing fibrations and p-harmonic maps in homotopy classes from S 3 to S 2 , Comm. Anal. Geom. 6 (1998), 427-483. Zbl0914.58010MR1638862
  30. [R2] T. Rivière, Dense subsets of H 1 / 2 ( S 2 , S 1 ) , Ann. Global Anal. Geom. 18 (2000), 517-528. Zbl0960.35022MR1790711
  31. [S] L. Simon, "Lectures on geometric measure theory", Proc. Centre for Math. Anal. 3, Australian National University, Canberra, 1983. Zbl0546.49019MR756417
  32. [SU] R. Schoen – K. Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps, J. Differential Geom. 18 (1983), 253-268. Zbl0547.58020MR710054
  33. [W1] B. White, Infima of energy functionals in homotopy classes of mappings, J. Differential Geom. 23 (1986), 127-142. Zbl0588.58017MR845702
  34. [W2] B. White, Homotopy classes in Sobolev spaces and the existence of energy minimizing maps, Acta Math. 160 (1988), 1-17. Zbl0647.58016MR926523
  35. [W3] B. White, Rectifiability of flat chains, Ann. Math. (2) 150 (1999), 165-184. Zbl0965.49024MR1715323
  36. [Z] Y. Zhou, "On the density of smooth maps in Sobolev spaces between two manifolds", Ph. D. thesis, Columbia University, 1993. 

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