On the existence and uniqueness of solutions of Möbius equations
HTML articles powered by AMS MathViewer
- by Xingwang Xu PDF
- Trans. Amer. Math. Soc. 337 (1993), 927-945 Request permission
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
- Thierry Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Springer-Verlag, New York, 1982. MR 681859, DOI 10.1007/978-1-4612-5734-9
- Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1963. MR 0152974 S. Kobayashi, Transformation groups of Riemannian manifolds, Interscience, New York, 1970.
- L. Zhiyong Gao and S.-T. Yau, The existence of negatively Ricci curved metrics on three-manifolds, Invent. Math. 85 (1986), no. 3, 637–652. MR 848687, DOI 10.1007/BF01390331
- Samuel I. Goldberg and Kentaro Yano, Manifolds admitting a non-homothetic conformal transformation, Duke Math. J. 37 (1970), 655–670. MR 268811
- Olli Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987. MR 867407, DOI 10.1007/978-1-4613-8652-0 A. Lichnerowicz, Géométrie des groupes de transformations, Dunod, Paris, 1958.
- Morio Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geometry 6 (1971/72), 247–258. MR 303464
- Morio Obata, Conformal transformations of compact Riemannian manifolds, Illinois J. Math. 6 (1962), 292–295. MR 138059
- Morio Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), 333–340. MR 142086, DOI 10.2969/jmsj/01430333
- Brad Osgood and Dennis Stowe, The Schwarzian derivative and conformal mapping of Riemannian manifolds, Duke Math. J. 67 (1992), no. 1, 57–99. MR 1174603, DOI 10.1215/S0012-7094-92-06704-4 —, A generalization of Nehari’s univalence criterion, preprint, 1988.
- Jean-Pierre Bourguignon and Jean-Pierre Ezin, Scalar curvature functions in a conformal class of metrics and conformal transformations, Trans. Amer. Math. Soc. 301 (1987), no. 2, 723–736. MR 882712, DOI 10.1090/S0002-9947-1987-0882712-7
- John M. Lee and Thomas H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 37–91. MR 888880, DOI 10.1090/S0273-0979-1987-15514-5
- Richard Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), no. 2, 479–495. MR 788292 R. Schoen and S. T. Yau, Differential geometry, vol. I, Beijing, 1988. (Chinese) M. Spivak, A comprehensive introduction to differential geometry, vol. 4, 2nd ed., Publish or Perish, Berkeley, Calif., 1979.
- Neil S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 265–274. MR 240748
- Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, vol. 94, Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1971 edition. MR 722297, DOI 10.1007/978-1-4757-1799-0
- Hidehiko Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), 21–37. MR 125546
- Kentaro Yano and Morio Obata, Conformal changes of Riemannian metrics, J. Differential Geometry 4 (1970), 53–72. MR 261500 S. T. Yau, Survey, Seminar On Differential Geometry (S. T. Yau, ed.), Princeton Univ. Press, 1982.
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 337 (1993), 927-945
- MSC: Primary 58G30; Secondary 53C20, 53C21
- DOI: https://doi.org/10.1090/S0002-9947-1993-1148047-5
- MathSciNet review: 1148047