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Generalized harmonic functions of Riemannian manifolds with ends

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Abstract

We study L-harmonic functions (solutions of the stationary Schrödinger equation) on arbitrary noncompact Riemannian manifolds with finitely many ends. We establish some existence and uniqueness results, and obtain sharp dimension estimates for L-harmonic functions on such manifolds.

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References

  1. Anderson M.T., Schoen R.: Positive harmonic functions on complete manifolds of negative curvature. Ann. Math. 121, 429–461 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  2. Grigor‘yan A.: Analitic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. 36, 135–249 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Grigor‘yan, A.: On the existence of positive fundamental solution of the Laplace equation on Riemannian manifolds. (in Russian). Matem. Sbornik. 128(3), 354–363 (1985). Engl. trans. Math USSR Sb. 56, 349–358 (1987)

  4. Grigor’yan, A.: Set of positive solutions of Laplace–Beltrami equation on special type of Riemannian manifolds. (in Russian). Izv. Vyssh. Uchebn. Zaved., Matematika. no. 2, 30–37 (1987). Engl. trans. Soviet Math (Iz. VUZ). 31(2), 48–60 (1987)

  5. Kim S.W., Lee Y.H.: Generalized Liouville property for Schrodinger operator on Riemannian manifolds. Math. Z. 238, 355–387 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Li P., Tam L.F.: Harmonic functions and the structure of complete manifolds. J. Differ. Geom. 35, 359–383 (1992)

    MathSciNet  MATH  Google Scholar 

  7. Losev A.G., Mazepa E.A.: Bounded solutions of the Shrödinger equation on Riemannian products. St. Petersburg Math. J. 13(1), 57–73 (2002)

    MathSciNet  MATH  Google Scholar 

  8. Losev A.G., Mazepa E.A., Chebanenko V.Y.: Unbounded solutions of the stationary Shrödinger equation on Riemannian manifolds. Comput. Methods Function Theory 3(2), 443–451 (2003)

    MathSciNet  MATH  Google Scholar 

  9. Mazepa E.A.: Boundary value problems for the stationary Schrödinger equation on Riemannian manifolds. Sib. Math. J. 43(3), 473–479 (2002)

    Article  MathSciNet  Google Scholar 

  10. Yau S.-T.: Harmonic function on complete Riemannian manifolds. Commun. Pure Appl. Math. 28, 201–228 (1975)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Information Technology, Volgograd State University, pr-t. Universitetsky, 100, Volgograd, 400062, Russia

    S. A. Korolkov & A. G. Losev

Authors
  1. S. A. Korolkov
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  2. A. G. Losev
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Correspondence to S. A. Korolkov.

Additional information

S. A. Korolkov and A. G. Losev are supported by the grant 10-01-97004-r_povolzh’ye_a from Russian Foundation for Basic Research.

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Korolkov, S.A., Losev, A.G. Generalized harmonic functions of Riemannian manifolds with ends. Math. Z. 272, 459–472 (2012). https://doi.org/10.1007/s00209-011-0943-2

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  • DOI: https://doi.org/10.1007/s00209-011-0943-2

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