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Existence of p-energy minimizers in homotopy classes and lifts of Newtonian maps

  • Elefterios SoultanisEmail author
Article

Abstract

We study the notion of p-quasihomotopy in Newtonian classes of mappings and link it to questions concerning lifts of Newtonian maps, under the assumption that the target space is nonpositively curved. Using this connection we prove that every p-quasihomotopy class of Newtonian maps contains a minimizer of the p-energy if the target has hyperbolic fundamental group.

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References

  1. [1]
    A. Björn and J. Björn, Nonlinear Potential Theory onMetric Spaces, EMS Tracts inMathematics, Vol. 17, European Mathematical Society (EMS), ZÜrich, 2011.CrossRefzbMATHGoogle Scholar
  2. [2]
    M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, in Grundlehren der Mathematischen Wissenschaften, Vol. 319, Springer-Verlag, Berlin, 1999.Google Scholar
  3. [3]
    N. Brodskiy, J. Dydak, B. Labuz, and A. Mitra, Covering maps for locally path-connected spaces, Fund. Math. 218 (2012), 13–46.Google Scholar
  4. [4]
    J. Eells, Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J.Math. 86 (1964), 109–160.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    M. Gromov and R. Schoen, Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one, Inst. Hautes Etudes Sci. Publ. Math. 76 (1992), 165–246.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    P. Hajłasz, Sobolev spaces on metric-measure spaces, in Heat Kernels and Analysis onManifolds, Graphs, and Metric Spaces (Paris, 2002), Contemp. Math., Vol. 338, American Mathematical Society, Providence, RI, 2003, pp. 173–218.Google Scholar
  7. [7]
    P. Hajłasz and P. Koskela, Sobolev met Poincaré, Mem. Amer.Math. Soc. 145 (2000).Google Scholar
  8. [8]
    R. Hardt and F.-H. Lin, Mappings minimizing the Lp norm of the gradient, Comm. Pure Appl. Math. 40 (1987), 555–588.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.zbMATHGoogle Scholar
  10. [10]
    J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer-Verlag, New York, 2001.CrossRefzbMATHGoogle Scholar
  11. [11]
    J. Heinonen, P. Koskela, N. Shanmugalingam, and J. Tyson, Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients, New Mathematical Monographs, Cambridge University Press, Cambridge, 2015.CrossRefzbMATHGoogle Scholar
  12. [12]
    J. Jost, Generalized harmonic maps between metric spaces, in Geometric Analysis and the Calculus of Variations, International Press, Cambridge, MA, 1996, pp. 143–174.Google Scholar
  13. [13]
    S. Kallunki and N. Shanmugalingam, Modulus and continuous capacity, Ann. Acad. Sci. Fenn. Math. 26 (2001), 455–464.MathSciNetzbMATHGoogle Scholar
  14. [14]
    S. Keith, Modulus and the Poincaré inequality on metric measure spaces, Math. Z. 245 (2003), 255–292.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    S. Keith and X. Zhong, The Poincaré inequality is an open ended condition, Ann. of Math. (2) 167 (2008), 575–599.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    N. J. Korevaar and R. M. Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993), 561–659.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    W. S. Massey, Algebraic Topology: An Introduction, Harcourt, Brace & World, Inc., New York, 1967.zbMATHGoogle Scholar
  18. [18]
    A. Papadopoulos, Metric Spaces, Convexity and Nonpositive Curvature, in IRMA Lectures in Mathematics and Theoretical Physics, Vol. 6, European Mathematical Society (EMS), ZÜrich, 2005.zbMATHGoogle Scholar
  19. [19]
    S. Pigola and G. Veronelli, On the homotopy class of maps with finite p-energy into non-positively curved manifolds, Geom. Dedicata 143 (2009), 109–116.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    R. Schoen and S. T. Yau, Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature, Comment. Math. Helv. 51 (1976), 333–341.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    R. Schoen and S. T. Yau, Compact group actions and the topology of manifolds with nonpositive curvature, Topology 18 (1979), 361–380.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    E. Soultanis, Homotopy classes of Newtonian spaces, Rev. Mat. Iberoamericana 33 (2017), 951–994.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    B. White, Homotopy classes in Sobolev spaces and the existence of energy minimizing maps, Acta Math. 160 (1988), 1–17.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    S. Willard, General Topology, Dover Publications, Inc., Mineola, NY, 2004.zbMATHGoogle Scholar

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© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.University of HelsinkiHelsinkiFinland

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