# zbMATH — the first resource for mathematics

Estimates by gap potentials of free homotopy decompositions of critical Sobolev maps. (English) Zbl 1437.58008
The free homotopy decomposition appears in the construction of harmonic and polyharmonic maps being known in many examples to generate all the homotopy classes through free homotopy decomposition (Theorem 14 in [A. Gastel and A. J. Nerf, Calc. Var. Partial Differ. Equ. 47, No. 3–4, 499–521 (2013; Zbl 1276.58003)], Theorem 5.5 in [J. Sacks and K. Uhlenbeck, Ann. Math. (2) 113, 1–24 (1981; Zbl 0462.58014)]). The free homotopy decomposition is an invariant under homotopies of the maps, but is not in general a faithful invariant. The author shows that maps of the same free homotopy decomposition abide by the same fractional Sobolev bound (Theorem 1.2).
Theorem. Let $$m\in\mathbb{N}$$ and $$\mathcal{N}$$ be a compact Riemannian manifold. If $$f\in\mathcal{C}(\mathbb{S}^{m},\mathcal{N})$$ has a free homotopy decomposition into $$f_{1},\dots,f_{k}\in\mathcal{C}(\mathbb{S}^{m},\mathcal{N})$$, then, for every $$s\in(0,1]$$ and every $$p\in\lbrack m,+\infty)$$ with $$p=m/s>1$$, one has $\inf\left\{ \mathcal{E}^{s,p}(g)\mid g\in(\mathcal{C}\cap W^{s,p})(\mathbb{S}^{m},\mathcal{N})\text{ is homotopic to }f\right\} \leq\sum_{i=1}^{k}\mathcal{E}^{s,p}(f_{i}).$
The proof of the above theorem is performed by gluing together the maps $$f_{1},\dots,f_{k}$$ with an arbitrarily small energetic cost of gluing through conformal transformations by Mercator projections.
By taking the above theorem into account, it was established that, for every $$\lambda>0$$, there exists a finite set $$\mathcal{F}$$ and $$k\in\mathbb{N}$$ such that every map $$f\in(\mathcal{C}\cap W^{s,p})(\mathbb{S}^{m},\mathcal{N})$$ pursuant to $$\mathcal{E}^{s,m/s}(f)\leq\lambda$$ has a free homotopy decomposition into $$k$$ maps of the set $$\mathcal{F}$$
in the case of $$m=1$$, $$s=\frac{1}{2}$$, $$p=2$$ [E. Kuwert, J. Reine Angew. Math. 505, 1–22 (1998; Zbl 0933.58014)]
in the case of $$m\geq1$$, $$s=1$$ [F. Duzaar and E. Kuwert, Calc. Var. Partial Differ. Equ. 6, No. 4, 285–313 (1998; Zbl 0909.49008), Theorem 4]
in the case of $$m\geq1$$, $$s=1-\frac{1}{m+1}$$ [T. Müller, Manuscr. Math. 103, No. 4, 513–540 (2000; Zbl 0981.49025), Theorem 5.1]
in the case of $$m=2$$, $$s=1$$ [R. Schoen and J. Wolfson, J. Differ. Geom. 58, No. 1, 1–86 (2001; Zbl 1052.53056), Lemma 5.2]

The critical case $$sp=m$$ for estimates is to be seen as a limiting case between the classical continuous picture of homotopy classes in the supercritical $$sp>m$$ and the combination of collapses and appearances of homotopy classes in the subcritical case $$sp<m$$ ([B. White, J. Differ. Geom. 23, 127–142 (1986; Zbl 0588.58017)], [H. Brezis and Y. Li, C. R. Acad. Sci., Paris, Sér. I, Math. 331, No. 5, 365–370 (2000; Zbl 0972.46014); J. Funct. Anal. 183, No. 2, 321–369 (2001; Zbl 1001.46019)], [F. Hang and F. Lin, Discrete Contin. Dyn. Syst. 13, No. 5, 1097–1124 (2005; Zbl 1093.46017); Acta Math. 191, No. 1, 55–107 (2003; Zbl 1061.46032); Commun. Pure Appl. Math. 56, No. 10, 1383–1415 (2003; Zbl 1038.46026); Math. Res. Lett. 8, No. 3, 321–330 (2001; Zbl 1049.46018)]).
The main result of the paper, claiming that the above estimates are in fact consequences of a stronger gap potential estimate, goes as follows:
Theorem. Let $$m\in\mathbb{N}$$ and $$\mathcal{N}$$ be a compact Riemannian manifold. If $$\varepsilon>0$$ is small enough, then there is a constant $$C>0$$ such that, for every $$\lambda>0$$, there exists a finite set $$\mathcal{F}^{\lambda}\subset\mathcal{C}(\mathbb{S}^{m},\mathcal{N})$$ with any $$f$$ pursuant to the inequality $\iint\limits_{\substack{(x,y)\in\mathbb{S}^{m}\times\mathbb{S}^{m}\\ d_{\mathcal{N}}(f(y),f(x))>\varepsilon}} \frac{1}{\left\vert y-x\right\vert ^{2m}}dydx\leq\lambda$ being of a free homotopy decomposition into $$f_{1},\dots,f_{k}$$ with $$k\leq C\lambda$$.
The above theorem describes sharply the homotopy classes that can be encountered under a boundedness assumption on the double integral. The proof of the above theorem goes within a geometric setting where the sphere $$\mathbb{S}^{m}$$ is put down as the boundary at infinity of the hyperbolic space $$\mathbb{H}^{m+1}$$, and the manifold $$\mathcal{N}$$ is embedded isometrically into a Euclidean space $$\mathbb{R}^{v}$$. The extension of the map $$f$$ by averaging at each point $$x\in\mathbb{H}^{m+1}$$ over the sphere at infinity provides a Lipschitz-continuous extension $$F:\mathbb{H}^{m+1}\rightarrow\mathbb{R}^{v}$$. The set on which the values of the map $$F$$ cannot be retracted to $$\mathcal{N}$$ is contained in a number of balls whose diameter and number is controlled with allowing to construct the families of maps by the classical Ascoli compactness argument for continuous maps.
The appearance of free homotopy decomposition in which the way of gluing the maps is uncontrolled is to be thought of as topological bubbling phenomenon, which is a topological version of the geometric bubbling phenomenon in conformally invariant geometric problems [O. Druet et al., Blow-up theory for elliptic PDEs in Riemannian geometry. Princeton, NJ: Princeton University Press (2004; Zbl 1059.58017); T. H. Parker, J. Differ. Geom. 44, No. 3, 595–633 (1996; Zbl 0874.58012); J. Sacks and K. Uhlenbeck, Ann. Math. (2) 113, 1–24 (1981; Zbl 0462.58014)]. In many cases, however, the above main result implies that maps abiding by a bound on the gap potential can only belong to finitely many homotopy classes (Theorem 1.4). In the one-dimensional case, the total variation of the maps occurring in the decomposition can be estimated (Theorem 1.5).

##### MSC:
 58C10 Holomorphic maps on manifolds 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 55M25 Degree, winding number 55P99 Homotopy theory 55Q25 Hopf invariants 58A12 de Rham theory in global analysis
Full Text:
##### References:
 [1] L. V. Ahlfors, Möbius transformations in several dimensions, Ordway Professorship Lectures in Mathematics, University of Minnesota School of Mathematics, Minneapolis, Minn., 1981. · Zbl 0517.30001 [2] H. J. Baues, Obstruction theory on homotopy classification of maps, Lecture Notes in Mathematics, vol. 628, Springer, Berlin-New York, 1977. · Zbl 0361.55017 [3] R. Bott and L. W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer, New York-Berlin, 1982. · Zbl 0496.55001 [4] J. Bourgain, H. Brezis, and P. Mironescu, Lifting, degree, and distributional Jacobian revisited, Comm. Pure Appl. Math. 58 (2005), no. 4, 529-551, . · Zbl 1077.46023 [5] J. Bourgain, H. Brezis, and Nguyen H.-M., A new estimate for the topological degree, C. R. Math. Acad. Sci. Paris 340 (2005), no. 11, 787-791, . · Zbl 1071.55002 [6] A. Boutet de Monvel-Berthier, V. Georgescu, and R. Purice, A boundary value problem related to the Ginzburg-Landau model, Comm. Math. Phys. 142 (1991), no. 1, 1-23. · Zbl 0742.35045 [7] H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, N.Y., 2011. · Zbl 1220.46002 [8] H. Brezis and Li Y. Y., Topology and Sobolev spaces, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), no. 5, 365-370, . · Zbl 0972.46014 [9] H. Brezis and Li Y. Y., Topology and Sobolev spaces, J. Funct. Anal. 183 (2001), no. 2, 321-369, . · Zbl 1001.46019 [10] H. Brezis and Nguyen H.-M., On a new class of functions related to VMO, C. R. Math. Acad. Sci. Paris 349 (2011), no. 3-4, 157-160, . · Zbl 1223.46027 [11] H. Brezis and L. Nirenberg, Degree theory and BMO. I: Compact manifolds without boundaries, Selecta Math. (N.S.) 1 (1995), no. 2, 197-263, . · Zbl 0852.58010 [12] Chen J. and Li Y., Homotopy classes of harmonic maps of the stratified 2-spheres and applications to geometric flows, Adv.Math. 263 (2014), 357-388, . · Zbl 1303.58005 [13] O. Druet, E. Hebey, and F. Robert, Blow-up theory for elliptic PDEs in Riemannian geometry, Mathematical Notes, vol. 45, Princeton University Press, Princeton, N.J., 2004. · Zbl 1059.58017 [14] F. Duzaar and E. Kuwert, Minimization of conformally invariant energies in homotopy classes, Calc. Var. Partial Differential Equations 6 (1998), no. 4, 285-313, . · Zbl 0909.49008 [15] B. Dyda, A fractional order Hardy inequality, Illinois J. Math. 48 (2004), no. 2, 575-588. · Zbl 1068.26014 [16] W. Fenchel, Elementary geometry in hyperbolic space, De Gruyter Studies in Mathematics, vol. 11, de Gruyter, Berlin, 1989. · Zbl 0674.51001 [17] W. H. Fleming, Flat chains over a finite coefficient group, Trans. Amer. Math. Soc. 121 (1966), 160-186, . · Zbl 0136.03602 [18] A. Gastel and A. J. Nerf, Minimizing sequences for conformally invariant integrals of higher order, Calc. Var. Partial Differential Equations 47 (2013), no. 3-4, 499-521, . · Zbl 1276.58003 [19] P. Goldstein and P. Hajłasz, Sobolev mappings, degree, homotopy classes and rational homology spheres, J. Geom. Anal. 22 (2012), no. 2, 320-338, . · Zbl 1262.58010 [20] M. Gromov, Quantitative homotopy theory, Prospects in mathematics (Princeton, N.J., 1996), Amer. Math. Soc., Providence, R.I., 1999, pp. 45-49. · Zbl 0927.57022 [21] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, translated by S. M. Bates, with appendices by M. Katz, P. Pansu, and S. Semmes, Progress in Mathematics, vol. 152, Birkhäuser, Boston, Mass., 1999. [22] P. Hajłasz, T. Iwaniec, J. Malý, and J. Onninen, Weakly differentiable mappings between manifolds, Mem. Amer. Math. Soc. 192 (2008), no. 899, . · Zbl 1145.58006 [23] Hang F. and Lin F., Topology of Sobolev mappings, Math. Res. Lett. 8 (2001), no. 3, 321-330, . · Zbl 1049.46018 [24] Hang F. and Lin F., Topology of Sobolev mappings. II, Acta Math. 191 (2003), no. 1, 55-107, . · Zbl 1061.46032 [25] Hang F. and Lin F., Topology of Sobolev mappings. III, Comm. Pure Appl. Math. 56 (2003), no. 10, 1383-1415, . · Zbl 1038.46026 [26] Hang F. and Lin F., Topology of Sobolev mappings. IV, Discrete Contin. Dyn. Syst. 13 (2005), no. 5, 1097-1124, . · Zbl 1093.46017 [27] A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. · Zbl 1044.55001 [28] S.-t. Hu, Homotopy theory, Pure and Applied Mathematics, Vol. VIII, Academic Press, New York-London, 1959. [29] R. L. Jerrard, Lower bounds for generalized Ginzburg-Landau functionals, SIAM J. Math. Anal. 30 (1999), no. 4, 721-746, . · Zbl 0928.35045 [30] E. Kuwert, A compactness result for loops with anH^1/2-bound, J. Reine Angew. Math. 505 (1998), 1-22, . · Zbl 0933.58014 [31] J. M. Lee, Manifolds and differential geometry, Graduate Studies in Mathematics, vol. 107, American Mathematical Society, Providence, R.I., 2009. · Zbl 1190.58001 [32] W. Magnus, A. Karrass, and D. Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers (John Wiley & Sons), New York-London-Sydney, 1966. · Zbl 0138.25604 [33] P. Mironescu, Sobolev maps on manifolds: degree, approximation, lifting, Perspectives in nonlinear partial differential equations, Contemp. Math., vol. 446, Amer. Math. Soc., Providence, R.I., 2007, pp. 413-436, . · Zbl 1201.46032 [34] T. Müller, Compactness for maps minimizing the n-energy under a free boundary constraint, Manuscripta Math. 103 (2000), no. 4, 513-540, . · Zbl 0981.49025 [35] Nguyen H.-M., Some new characterizations of Sobolev spaces, J. Funct. Anal. 237 (2006), no. 2, 689-720, . · Zbl 1109.46040 [36] Nguyen H.-M., Γ-convergence and Sobolev norms, C. R. Math. Acad. Sci. Paris 345 (2007), no. 12, 679-684, . · Zbl 1132.46026 [37] Nguyen H.-M., Optimal constant in a new estimate for the degree, J. Anal. Math. 101 (2007), 367-395, . · Zbl 1147.47046 [38] Nguyen H.-M., Further characterizations of Sobolev spaces, J. Eur. Math. Soc. (JEMS) 10 (2008), no. 1, 191-229, . · Zbl 1228.46033 [39] H.-M. Nguyen, Inequalities related to liftings and applications, C. R. Math. Acad. Sci. Paris 346 (2008), no. 17-18, 957-962, . · Zbl 1157.46016 [40] Nguyen H.-M., Γ-convergence, Sobolev norms, and BV functions, Duke Math. J. 157 (2011), no. 3, 495-533, . · Zbl 1221.28011 [41] H.-M. Nguyen, Some inequalities related to Sobolev norms, Calc. Var. Partial Differential Equations 41 (2011), no. 3-4, 483-509, . · Zbl 1226.46030 [42] Nguyen H.-M., Estimates for the topological degree and related topics, J. Fixed Point Theory Appl. 15 (2014), no. 1, 185-215, . · Zbl 1321.46037 [43] Nguyen H.-M., A refined estimate for the topological degree, C. R. Math. Acad. Sci. Paris 355 (2017), no. 10, 1046-1049, . · Zbl 1422.55004 [44] H.-M. Nguyen, A. Pinamonti, M. Squassina, and E. Vecchi, New characterizations of magnetic Sobolev spaces, Adv. Nonlinear Anal. 7 (2018), no. 2, 227-245, . · Zbl 1429.49007 [45] T. H. Parker, What is... a bubble tree?, Notices Amer. Math. Soc. 50 (2003), no. 6, 666-667. [46] P. Petersen, Riemannian geometry, 3rd ed., Graduate Texts in Mathematics, vol. 171, Springer, Cham, 2016. · Zbl 1417.53001 [47] M. Petrache and T. Rivière, Global gauges and global extensions in optimal spaces, Anal. PDE 7 (2014), no. 8, 1851-1899, . · Zbl 1328.46034 [48] M. Petrache and J. Van Schaftingen, Controlled singular extension of critical trace Sobolev maps from spheres to compact manifolds, Int. Math. Res. Not. IMRN 12 (2017), 3467-3683, . · Zbl 1405.58002 [49] T. Rivière, Minimizing fibrations and p-harmonic maps in homotopy classes from S3 into S2, Comm. Anal. Geom. 6 (1998), no. 3, 427-483, . · Zbl 0914.58010 [50] J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1-24, . · Zbl 0462.58014 [51] E. Sandier, Lower bounds for the energy of unit vector fields and applications, J. Funct. Anal. 152 (1998), no. 2, 379-403, . · Zbl 0908.58004 [52] E. Sandier and S. Serfaty, Vortices in the magnetic Ginzburg-Landau model, Progress in Nonlinear Differential Equations and their Applications, vol. 70, Birkhäuser, Boston, Mass., 2007. · Zbl 1112.35002 [53] A. Schikorra and J. Van Schaftingen, An estimate of the Hopf degree of fractional Sobolev mappings. arXiv:1904.12549. [54] R. Schoen and J. Wolfson, Minimizing area among Lagrangian surfaces: the mapping problem, J. Differential Geom. 58 (2001), no. 1, 1-86, . · Zbl 1052.53056 [55] J.-P. Serre, Homologie singulière des espaces fibrés. Applications, Ann. of Math. (2) 54 (1951), 425-505, . · Zbl 0045.26003 [56] Toda H., Composition methods in homotopy groups of spheres, Annals of Mathematics Studies, No. 49, Princeton University Press, Princeton, N.J., 1962. · Zbl 0101.40703 [57] B. White, Infima of energy functionals in homotopy classes of mappings, J. Differential Geom. 23 (1986), no. 2, 127-142, . · Zbl 0588.58017 [58] J. H. C. Whitehead, An expression of Hopf’s invariant as an integral, Proc. Nat. Acad. Sci. U. S. A. 33 (1947), 117-123. · Zbl 0030.07902 [59] M. Willem, Functional analysis: Fundamentals and applications, Cornerstones, Birkhäuser/Springer, New York, N.Y., 2013.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.