Connecting topological Hopf singularities

Hardt, Robert; Rivière, Tristan

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 2, pp. 287-344.

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 ${\pi }_2\left(S^2\right)$ 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 ${\pi }_3\left(S^2\right)$ 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.

[AK] L. Ambrosio - B. Kirchheim, Currents in metric spaces, Acta Math. 185 (2000), 1-80. | MR | Zbl

[Be1] F. Bethuel, The approximation problem for Sobolev maps between two manifolds, Acta Math. 167 (1991), 153-206. | MR | Zbl

[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. | Numdam | MR | Zbl

[Be3] F. Bethuel, Approximations in trace spaces defined between manifolds, Nonlinear Anal. 24 (1995), 121-130. | MR | Zbl

[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. | MR | Zbl

[BCL] H. Brezis - J.-M. Coron, - E. Lieb, Harmonic maps with defects, Comm. Math. Phys. 107, no. 4 (1986), 649-705. | MR | Zbl

[BL] H. Brezis - Y.Y. Li, Topology and Sobolev spaces, preprint, 2000. | MR | Zbl

[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. | MR | Zbl

[F] H. Federer, "Geometric measure theory", Springer-Verlag, Berlin and New York, 1969. | MR | Zbl

[Fl] Fleming, flat chains over a coefficient group, Trans. Amer. Math. Soc. 121 (1966), 160-186. | MR | Zbl

[G] M. Giaquinta, “Multiple integrals in the calculus of variations and elliptic systems”, Princeton University Press, Princeton, 1983. | MR | Zbl

[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. | MR | Zbl

[GMS2] M. Giaquinta - G. Modica - J. Souček, "Cartesian currents in the calculus of variations". I, II, Springer-Verlag, Berlin, 1998. | MR | Zbl

[Gr] M. Gromov, "Metric structures for Riemannian and non-Riemannian spaces", Progress in mathematics 152, Birkhaüser, Boston, 1999. | MR | Zbl

[H] R. Hardt, Uniqueness of nonparametric area minimizing currents, Indiana Univ. Math. J. 26 (1977), 65-71. | MR | Zbl

[HD] R. Hardt - T. Depauw, Size minimization and approximating problems, to appear in Calc. Var. Partial Differential Equations. | MR | Zbl

[He] F. Hélein, "Harmonic maps, conservation laws, and moving frames", Diderot Press, Paris, 1997. | MR | Zbl

[HgL1] F. Hang - Lin, Topology of Sobolev mappings, Math. Res. Lett. 8 (2001), 321-330. | MR | Zbl

[HgL2] F. Hang - F. Lin, Topology of Sobolev mappings II, preprint, 2001. | MR | Zbl

[HL1] R. Hardt - F.H. Lin, A remark on $H^1$ mappings, Manuscripta Math. 56 (1986), 1-10. | MR | Zbl

[HL2] R. Hardt - F.H. Lin, Mappings minimizing the $L^p$ norm of the gradient, Comm. Pure. Appl. Math. 15 (1987), 555-588. | MR | Zbl

[HR1] R. Hardt - T. Rivière, Ensembles singuliers topologiques dans les espaces fonctionnels entre variétés, Exp. VII, Sémin. Équ. Dériv. Part., École Polytech., Palaiseau, 2001. | Numdam | MR | Zbl

[HR2] R. Hardt - T. Rivière, Bubbling phenomena for maps in $W^{2,2}({𝔹}^5,{\S }^3)$, in preparation.

[HR3] R. Hardt - T. Rivière, Connecting singularities of arbitrary rational homotopy type, in preparation.

[I1] T. Isobe, Characterization of the strong closure of $C^{\infty }(B^4,S^2)$ in $W^{1,p}(B^4,S^2)$ $(16/5\le p<4)$, J. Math. Anal. Appl. 190 (1995), 361-372. | MR | Zbl

[I2] T. Isobe, Some new properties of Sobolev mappings: intersection theoretical approach, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), 337-358. | MR | Zbl

[M] D. Mucci, A characterization of graphs which can be approximated in area by smooth graphs, To appear in the J. Eur. Math. Soc. | MR | Zbl

[PR] M. R. Pakzad - T. Rivière, Weak density of smooth maps for the Dirichlet energy between manifolds, preprint, 2000. | MR | Zbl

[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. | MR | Zbl

[R2] T. Rivière, Dense subsets of $H^{1/2}(S^2,S^1)$, Ann. Global Anal. Geom. 18 (2000), 517-528. | MR | Zbl

[S] L. Simon, "Lectures on geometric measure theory", Proc. Centre for Math. Anal. 3, Australian National University, Canberra, 1983. | MR | Zbl

[SU] R. Schoen - K. Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps, J. Differential Geom. 18 (1983), 253-268. | MR | Zbl

[W1] B. White, Infima of energy functionals in homotopy classes of mappings, J. Differential Geom. 23 (1986), 127-142. | MR | Zbl

[W2] B. White, Homotopy classes in Sobolev spaces and the existence of energy minimizing maps, Acta Math. 160 (1988), 1-17. | MR | Zbl

[W3] B. White, Rectifiability of flat chains, Ann. Math. (2) 150 (1999), 165-184. | MR | Zbl

[Z] Y. Zhou, "On the density of smooth maps in Sobolev spaces between two manifolds", Ph. D. thesis, Columbia University, 1993.