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DEFORMING SOLUTIONS OF GEOMETRIC VARIATIONAL PROBLEMS WITH VARYING SYMMETRY GROUPS

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Abstract

We prove an equivariant implicit function theorem for variational problems that are invariant under a varying symmetry group (corresponding to a bundle of Lie groups). Motivated by applications to families of geometric variational problems lacking regularity, several non-smooth extensions of the result are discussed. Among such applications is the submanifold problem of deforming the ambient metric preserving a given variational property of a prescribed family of submanifolds, e.g., constant mean curvature, up to the action of the corresponding ambient isometry groups.

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References

  1. L. J. Alías, P. Piccione, On the manifold structure of the set of unparametrized embeddings with low regularity, Bull. Braz. Math. Soc. (N.S.) 42 (2011), no. 2, 171–183.

    Article  MATH  MathSciNet  Google Scholar 

  2. R. G. Bettiol, P. Piccione, G. Siciliano, On the equivariant implicit function theorem with low regularity and applications to geometric variational problems, to appear in Proc. Edinb. Math. Soc., arXiv:1009.5721.

  3. R. G. Bettiol, P. Piccione, B. Santoro, Equivariant deformations of Hamiltonian stationary Lagrangian submanifolds, to appear in Mat. Contemp., arXiv:1302.6970.

  4. V. Cervera, F. Mascaró, P. W. Michor, The action of the diffeomorphism group on the space of immersions, Diff. Geom. Appl. 1 (1991), 391–401.

    Article  MATH  Google Scholar 

  5. D. Coppersmith, Deformations of Lie subgroups, Trans. Amer. Math. Soc. 233 (1977), 355–366.

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Crainic, R. L. Fernandes, Integrability of Lie brackets, Ann. of Math. (2) 157 (2003), no. 2, 575–620.

  7. E. N. Dancer, The G-invariant implicit function theorem in infinite dimension, Proc. Roy. Soc. Edinburgh Sect. A 92 (1982), no. 1–2, 13–30.

    Article  MATH  MathSciNet  Google Scholar 

  8. E. N. Dancer, The G-invariant implicit function theorem in infinite dimension. II., Proc. Roy. Soc. Edinburgh Sect. A 102 (1986), no. 3–4, 211–220.

    Article  MATH  MathSciNet  Google Scholar 

  9. B. Daniel, Isometric immersions into \( {S^n}\times \mathbb{R} \) and \( {H^n}\times \mathbb{R} \) and applications to minimal surfaces, Trans. Amer. Math. Soc. 361 (2009), no. 12, 6255–6282.

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Douady, M. Lazard, Espaces fibrés en algèbres de Lie et en groupes, Invent. Math. 1 (1966) 133–151.

    Article  MATH  MathSciNet  Google Scholar 

  11. D. Hoffman, B. White, Axial minimal surfaces in \( {S^2}\times \mathbb{R} \) are helicoidal, J. Diff. Geom. 87 (2011), no. 3, 515–523.

    MATH  MathSciNet  Google Scholar 

  12. D. Hoffman, M. Traizet, B. White, Helicoidal minimal surfaces of prescribed genus, I, II, arXiv:1304.5861, arXiv:1304.6180.

  13. D. Joyce, Y.-I. Lee, R. Schoen, On the existence of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds, Amer. J. Math. 133 (2011), no. 4, 1067–1092.

    Article  MATH  MathSciNet  Google Scholar 

  14. L. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, London Mathematical Society Lecture Note Series, Vol. 124, Cambridge University Press, Cambridge, 1987.

  15. W. Meeks, H. Rosenberg, The theory of minimal surfaces in \( M\times \mathbb{R} \), Comment. Math. Helv. 80 (2005), no. 4, 811–858.

    Article  MATH  MathSciNet  Google Scholar 

  16. I. Moerdijk, On the classification of Regular Lie groupoids, preprint no. 1228, Department of Mathematics, Utrecht University, 2002.

  17. I. Moerdijk, J. Mrčun, Introduction to Foliations and Lie Groupoids. Cambridge Studies in Advanced Mathematics, Vol. 91, Cambridge University Press, Cambridge, 2003.

  18. I. Moerdijk, J. Mrčun, On integrability of infinitesimal actions, Amer. J. Math. 124 (2002), no. 3, 567–593.

    Article  MATH  MathSciNet  Google Scholar 

  19. I. Moerdijk, J. Mrčun, On the integrability of Lie subalgebroids, Adv. Math. 204 (2006), no. 1, 101–115.

    Article  MATH  MathSciNet  Google Scholar 

  20. F. Morabito, A Costa–Hoffman–Meeks type surface in \( {{\mathbb{H}}^2}\times \mathbb{R} \), Trans. Amer. Math. Soc. 363 (2011), 1–36.

    Article  MATH  MathSciNet  Google Scholar 

  21. R. S. Palais, Foundations of Global Non-linear Analysis, W. A. Benjamin, New York, 1968.

    MATH  Google Scholar 

  22. P. Piccione, D. V. Tausk, On the Banach differential structure for sets of maps on non-compact domains, Nonlinear Anal. 46 (2001), no. 2, Ser. A: Theory Methods, 245–265.

  23. R. W. Richardson, Jr., Deformations of Lie subgroups and the variation of isotropy subgroups, Acta Math. 129 (1972), 35–73.

    Article  MATH  MathSciNet  Google Scholar 

  24. H. Tasaki, M. Umehara, K. Yamada, Deformations of symmetric spaces of compact type to their noncompact duals, Japan J. Math. (N.S.) 17 (1991), no. 2, 383–399.

    MATH  MathSciNet  Google Scholar 

  25. M. Umehara, K. Yamada, A deformation of tori with constant mean curvature in \( {{\mathbb{R}}^3} \) to those in other space forms, Trans. AMS 330, no. 2 (1992), 845–857.

    MATH  MathSciNet  Google Scholar 

  26. A. Weinstein, Linearization of regular proper groupoids, J. Inst. Math. Jussieu 1 (2002), no. 3, 493–511.

    Article  MATH  MathSciNet  Google Scholar 

  27. H. Wente, Counterexample to a conjecture of H. Hopf, Pacific J. Math. 121 (1986), no. 1, 193–243.

    Article  MATH  MathSciNet  Google Scholar 

  28. B. White, The space of minimal submanifolds for varying Riemannian metrics, Indiana Univ. Math. J. 40 (1991), 161–200.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame, 255 Hurley Building, Notre Dame, IN, 46556-4618, USA

    R. G. BETTIOL

  2. Departamento de Matemática, Universidade de São Paulo, Rua do Matão, 1010, São Paulo, SP, 05508-090, Brazil

    P. PICCIONE & G. SICILIANO

Authors
  1. R. G. BETTIOL
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  2. P. PICCIONE
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  3. G. SICILIANO
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Corresponding author

Correspondence to R. G. BETTIOL.

Additional information

*Supported by the NSF grants DMS-0941615 and DMS-1209387, USA.

**Partially supported by Fapesp and CNPq, Brazil.

***Partially supported by Fapesp and CNPq, Brazil.

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BETTIOL, R.G., PICCIONE, P. & SICILIANO, G. DEFORMING SOLUTIONS OF GEOMETRIC VARIATIONAL PROBLEMS WITH VARYING SYMMETRY GROUPS. Transformation Groups 19, 941–968 (2014). https://doi.org/10.1007/s00031-014-9277-6

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  • DOI: https://doi.org/10.1007/s00031-014-9277-6

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