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Transactions of the American Mathematical Society
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
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1148047-5
PII: S 0002-9947(1993)1148047-5
Article copyright: © Copyright 1993 American Mathematical Society