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The existence of generalized isothermal coordinates for higher-dimensional Riemannian manifolds

Author: Jian Guo Cao
Journal: Trans. Amer. Math. Soc. 324 (1991), 901-920
MSC: Primary 53B20; Secondary 53A30
MathSciNet review: 991959
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Abstract: We shall show that, for any given point $ p$ on a Riemannian manifold $ (M,{g^0})$, there is a pointwise conformal metric $ g = \Phi {g^0}$ in which the $ g$-geodesic sphere centered at $ p$ with radius $ r$ has constant mean curvature $ 1/r$ for all sufficiently small $ r$. Furthermore, the exponential map of $ g$ at $ p$ is a measure preserving map in a small ball around $ p$.

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  • [A] T. Aubin, Nonlinear analysis on manifolds. Monge-Ampere equations, Springer-Verlag, 1982. MR 681859 (85j:58002)
  • [BE] A. Besse, Einstein manifolds, Springer-Verlag, 1986. MR 867684 (88f:53087)
  • [BGE] M. Berger, P. Gauduchon, and E. Mazet, Le spectre d'une variete Riemannienne, Springer-Verlag, 1971. MR 0282313 (43:8025)
  • [CHS] L. Caffarelli, R. Hardt, and L. Simon, Minimal surfaces and isolated singularities, Manuscripta Math. 48 (1984), 1-18. MR 753722 (86h:53007)
  • [D] D. DeTurck, Existence of metrics with prescribed Ricci curvature, Invent. Math. 65 (1985), 197-207. MR 636886 (83b:53019)
  • [DK] D. DeTurck and J. Kazdan, Some regularity theorem in Riemannian geometry, Ann. Sci. École Norm. Sup. (4) 14 (1981), 249-260. MR 644518 (83f:53018)
  • [EH] J. Eschenberg and E. Heintze, An elementary proof of Cheeger-Gromoll splitting theorem, Global Analysis Geometry 2 (1984), 141-167. MR 777905 (86h:53042)
  • [GT] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equation of second order, Springer-Verlag, 1983. MR 737190 (86c:35035)
  • [GG] M. Golubitsky and V. Guillemen, Stable mapping and their singularities, Springer-Verlag, 1973. MR 0341518 (49:6269)
  • [GV] A. Gray and L. Vanhecke, Riemannian geometry and its determined by the volume of small geodesic balls, Acta Math. 142 (1979), 158-198. MR 521460 (81i:53038)
  • [LP] J. Lee and T. Parker, Yamabe problem, Bull. Amer. Math. Soc. (N.S.) 17 (1987), 37-91. MR 888880 (88f:53001)
  • [M] J. Milnor, Morse theory, Princeton Univ. Press, Princeton, N.J., 1963. MR 0163331 (29:634)

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Article copyright: © Copyright 1991 American Mathematical Society

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