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)



On the existence and uniqueness of solutions of Möbius equations

Author: Xingwang Xu
Journal: Trans. Amer. Math. Soc. 337 (1993), 927-945
MSC: Primary 58G30; Secondary 53C20, 53C21
MathSciNet review: 1148047
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A generalization of the Schwarzian derivative to conformal mappings of Riemannian manifolds has naturally introduced the corresponding overdetermined differential equation which we call the Möbius equation. We are interested in study of the existence and uniqueness of the solution of the Möbius equation. Among other things, we show that, for a compact manifold, if Ricci curvature is nonpositive, for a complete noncompact manifold, if the scalar curvature is a positive constant, then the differential equation has only constant solutions. We also study the nonhomogeneous equation in an $ n$-dimensional Euclidean space.

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

  • [1] T. Aubin, Nonlinear analysis on manifolds, Monge-Ampere equations, Springer-Verlag, 1982. MR 681859 (85j:58002)
  • [2] S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol. I, Interscience, New York, 1963. MR 0152974 (27:2945)
  • [3] S. Kobayashi, Transformation groups of Riemannian manifolds, Interscience, New York, 1970.
  • [4] L. Zhiyong Gao and S. T. Yau, The existence of negatively Ricci curved metrics on three manifolds, Invent. Math. 85 (1986), 637-652. MR 848687 (87j:53061)
  • [5] S. I. Goldberg and K. Yano, Manifolds admitting a non-homothetic conformal transformation, Duke Math. J. 37 (1970), 655-670. MR 0268811 (42:3708)
  • [6] O. Lehto, Univalent functions and Teichmüller spaces, Springer-Verlag, 1987. MR 867407 (88f:30073)
  • [7] A. Lichnerowicz, Géométrie des groupes de transformations, Dunod, Paris, 1958.
  • [8] M. Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geom. 6 (1971), 247-258. MR 0303464 (46:2601)
  • [9] -, Conformal transformations of compact Riemannian manifolds, Illinois J. Math. 6 (1962), 292-295. MR 0138059 (25:1507)
  • [10] -, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), 333-340. MR 0142086 (25:5479)
  • [11] B. Osgood and D. Stowe, The Schwarzian derivative and conformal mapping of Riemannian manifolds, preprint, 1987. MR 1174603 (93j:53062)
  • [12] -, A generalization of Nehari's univalence criterion, preprint, 1988.
  • [13] J. P. Bourguignon and J. P. Ezin, Scalar curvature functions in a conformal class of metrics and conformal transformations, Trans. Amer. Math. Soc. 301 (1987), 723-736. MR 882712 (88e:53054)
  • [14] J. M. Lee and Th. H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.) 17 (1987), 37-91. MR 888880 (88f:53001)
  • [15] R. Schoen, Conformal deformation of a riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), 479-495. MR 788292 (86i:58137)
  • [16] R. Schoen and S. T. Yau, Differential geometry, vol. I, Beijing, 1988. (Chinese)
  • [17] M. Spivak, A comprehensive introduction to differential geometry, vol. 4, 2nd ed., Publish or Perish, Berkeley, Calif., 1979.
  • [18] N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa 22 (1968), 265-274. MR 0240748 (39:2093)
  • [19] F. Warner, Foundations of differential manifolds and Lie groups, Springer-Verlag, 1983. MR 722297 (84k:58001)
  • [20] H. Yamabe, On the deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), 21-37. MR 0125546 (23:A2847)
  • [21] K. Yano and M. Obata, Conformal changes of Riemannian metrics, J. Differential Geom. 4 (1970), 53-92. MR 0261500 (41:6113)
  • [22] S. T. Yau, Survey, Seminar On Differential Geometry (S. T. Yau, ed.), Princeton Univ. Press, 1982.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58G30, 53C20, 53C21

Retrieve articles in all journals with MSC: 58G30, 53C20, 53C21

Additional Information

Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society