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Rigidity of vector valued harmonic maps of linear growth

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Abstract

Consider vector valued harmonic maps of at most linear growth, defined on a complete non-compact Riemannian manifold with non-negative Ricci curvature. For the square of the Jacobian of such maps, we report a strong maximum principle, and equalities among its supremum, its asymptotic average, and its large-time heat evolution.

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References

  1. Cheeger, J.: Degeneration of Riemannian metrics under Ricci curvature bounds (Lezioni Fermiane). Scuola Normale Superiore, Pisa (2001)

  2. Cheeger, J., Colding, T.: Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann. Math. 144, 189–237 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cheeger, J., Colding, T.: On the structure of spaces with Ricci curvature bounded below. I. J. Differ. Geom. 46, 406–480 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cheeger, J., Colding, T.: On the structure of spaces with Ricci curvature bounded below. II. J. Differ. Geom. 54, 1–35 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cheeger, J., Colding, T.: On the structure of spaces with Ricci curvature bounded below. III. J. Differ. Geom. 54, 37–74 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cheeger, J., Colding, T., Minicozzi II, W.: Linear growth harmonic functions on complete manifolds with non-negative Ricci curvature. Geom. Funct. Anal. 6, 948–954 (1995)

    Article  MATH  Google Scholar 

  7. Cheeger, J., Gromoll, D.: The splitting theorem for manifolds of nonnegative Ricci curvature. J. Differ. Geom. 6, 119–128 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cheeger, J., Naber, A.: Regularity of Einstein manifolds and the codimension \(4\) conjecture. Ann. Math. 182, 1093–1165 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cheng, S.-Y., Yau, S.-T.: Differential equations on Riemannian manifolds and their geometric applications. Commun. Pure Appl. Math. 28, 333–354 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  10. Colding, T., Minicozzi II, W.: Harmonic functions with polynomial growth. J. Differ. Geom. 46, 1–77 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Colding, T., Minicozzi II, W.: Harmonic functions on manifolds. Ann. Math. 146, 725–747 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Colding, T., Naber, A.: Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications. Ann. Math. 176, 1173–1229 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Donnelly, H., Fefferman, C.: Nodal domains and growth of harmonic functions on noncompact manifolds. J. Geom. Anal. 2, 79–93 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, P.: Uniqueness of \(L^1\) solutions for the Laplace equation and the heat equation on Riemannian manifolds. J. Differ. Geom. 20, 447–457 (1984)

    Article  MATH  Google Scholar 

  15. Li, P.: Large time behavior of the heat equation on complete manifolds with non-negative Ricci curvature. Ann. Math. 124, 1–21 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, P., Tam, L.-F.: Linear growth harmonic functions on a complete manifold. J. Differ. Geom. 29, 421–425 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, P., Tam, L.-F.: Complete surfaces with finite total curvature. J. Differ. Geom. 33, 139–168 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, P., Schoen, R.: \(L^p\) and mean value properties of subharmonic functions on Riemannian manifolds. Acta Math. 153, 279–301 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, P., Wang, J.-P.: Convex hull properties of harmonic maps. J. Differ. Geom. 48, 497–530 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, P., Yau, S.-T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986)

    Article  MathSciNet  Google Scholar 

  21. Saloff-Coste, L.: A note on Poincaré, Sobolev and Harnack inequalities. Int. Math. Res. Not. 2, 27–38 (1992)

    Article  MATH  Google Scholar 

  22. Schoen, R., Yau, S.-T.: Lectures on Differential Geometry. International Press, Boston (1994)

    MATH  Google Scholar 

  23. Xu, G.: Large time behavior of the heat kernel. J. Differ. Geom. 98, 467–528 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yau, S.-T.: Harmonic functions on a complete Riemannian manifold. Commun. Pure Appl. Math. 28, 201–228 (1975)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Shaosai Huang.

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Both authors are partially supported by NSF Grant DMS-1510401.

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Huang, S., Wang, B. Rigidity of vector valued harmonic maps of linear growth. Geom Dedicata 202, 357–371 (2019). https://doi.org/10.1007/s10711-018-0418-2

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  • DOI: https://doi.org/10.1007/s10711-018-0418-2

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