Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Asymptotic Dirichlet problems for harmonic functions on Riemannian manifolds


Author: Hyeong In Choi
Journal: Trans. Amer. Math. Soc. 281 (1984), 691-716
MSC: Primary 53C20; Secondary 31C12, 58G99
DOI: https://doi.org/10.1090/S0002-9947-1984-0722769-4
MathSciNet review: 722769
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We define the asymptotic Dirichlet problem and give a sufficient condition for solving it. This proves an existence of nontrivial bounded harmonic functions on certain classes of noncompact complete Riemannian manifolds.


References [Enhancements On Off] (What's this?)

  • [Al] L. Ahlfors, Sur le type d'une surface de Riemann, C. R. Acad. Sci. Paris 201 (1935), 30-32.
  • [An] M. Anderson, The Dirichlet problem at infinity for manifolds of negative curvature, preprint. MR 730923 (85m:58178)
  • [AS] L. Ahlfors and L. Sario, Riemann surfaces, Princeton Univ. Press, Princeton, N. J., 1960. MR 0114911 (22:5729)
  • [Ax] S. Alexander, Locally convex hypersurfaces of negatively curved spaces, Proc. Amer. Math. Soc. 64 (1977), 321-325. MR 0448262 (56:6571)
  • [B] R. Bishop, Infinitesimal convexity implies local convexity, Indiana Univ. Math. J. 24 (1974), 169-172. MR 0350662 (50:3154)
  • [Ch] S.-Y. Cheng, Liouville theorem for harmonic maps, Proc. Sympos. Pure Math., vol. 36, Amer. Math. Soc., Providence, R. I., 1980, pp. 147-151. MR 573431 (81i:58021)
  • [Ci1] H. I. Choi, On the Liouville theorem for harmonic maps, Proc. Amer. Math. Soc. 85 (1982), 91-94. MR 647905 (83j:53073)
  • [Ci2] -, Characterizations of simply connected rotationally symmetric manifolds, Trans. Amer. Math. Soc. 275 (1983), 723-727. MR 682727 (84c:53042)
  • [Ci3] -, Thesis, Univ. of California, Berkeley, June 1982.
  • [E1] P. Eberlein, Geodesics and ends in certain surfaces without conjugate points, Mem. Amer. Math. Soc. vol. 13, no. 199(1978). MR 482449 (82e:53070)
  • [E2] -, Surfaces of nonpositive curvature, Mem. Amer. Math. Soc. vol. 20, no. 218 (1979). MR 533654 (80j:53044)
  • [EO] P. Eberlein and B. O'Neill, Visibility manifolds. Pacific J. Math. 46 (1973), 45-109. MR 0336648 (49:1421)
  • [F] R. Finn, On a class of conformal metrics with applications to differential geometry in the large, Comment. Math. Helv. 40 (1965), 1-30. MR 0203618 (34:3467)
  • [GT] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin and New York, 1977. MR 0473443 (57:13109)
  • [GW1] R. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Math., vol. 699, Springer-Verlag, Berlin and New York, 1979. MR 521983 (81a:53002)
  • [GW2] -, On a new gap phenomenon in Riemannian geometry, Proc. Nat. Acad. Sci. U.S.A. (1982). MR 648065 (83b:53040)
  • [GKM] D. Gromoll, W. Klingenberg and W. Meyer, Riemannsche Geometrie im Grossen, Lecture Notes in Math., vol. 55, Springer-Verlag, Berlin and New York, 1975. MR 0365399 (51:1651)
  • [HJW] S. Hildebrandt, J. Jost and K.-O. Widman, Harmonic mappings and minimal submanifolds, Invent. Math. 62 (1980), 269-298. MR 595589 (82d:58025)
  • [H] A. Huber, On subharmonic functions and differential geometry in the large, Comment. Math. Helv. 32 (1957), 13-72. MR 0094452 (20:970)
  • [Ma] A. Markushevich, Theory of functions of a complex variable, vol. 2, Chelsea, New York, 1967.
  • [Mi] J. Milnor, On deciding whether a surface is parabolic or hyperbolic, Amer. Math. Monthly 84 (1977), 43-46. MR 0428232 (55:1257)
  • [S] D. Sullivan, The Dirichlet problem at infinity for a negatively curved manifold, preprint. MR 730924 (85m:58177)
  • [W] H. Wu, Normal families of holomorphic mappings, Acta Math. 119 (1967), 193-233. MR 0224869 (37:468)
  • [Ya] P. Yang, Curvature of complex submanifolds of $ {{\mathbf{C}}^n}$ J. Differential Geom. 12 (1977), 499-511. MR 512921 (80c:53071)
  • [Yu] S.- T. Yau, Harmonic functions on complete Riemannian manifolds, Com. Pure Appl. Math. 28 (1975), 201-228. MR 0431040 (55:4042)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 53C20, 31C12, 58G99

Retrieve articles in all journals with MSC: 53C20, 31C12, 58G99


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0722769-4
Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society