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Surfaces of revolution with monotonic increasing curvature and an application to the equation $ \Delta u=1-K e^{2u}$ on $ S^{2}$


Authors: Jerry L. Kazdan and Frank W. Warner
Journal: Proc. Amer. Math. Soc. 32 (1972), 139-141
MSC: Primary 53.75; Secondary 35.00
DOI: https://doi.org/10.1090/S0002-9939-1972-0290309-X
MathSciNet review: 0290309
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Abstract: The geometric result that a compact surface of revolution in $ {R^3}$ cannot have monotonic increasing curvature is proved and applied to show that the equation $ \Delta u = 1 - K{e^{2u}}$, on $ {S^2}$, has no axially symmetric solutions u, given axially symmetric data K.


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

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  • [2] J. Kazdan and F. Warner, Integrability conditions for $ \Delta u = k - K{e^{\alpha u}}$ with applications to Riemannian geometry, Bull. Amer. Math. Soc. 77 (1971), 819-823. MR 0282314 (43:8026)
  • [3] L. Nirenberg, The Weyland Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math. 6 (1953), 337-394. MR 15, 347. MR 0058265 (15:347b)
  • [4] B. O'Neill, Elementary differential geometry, Academic Press, New York, 1966. MR 34 #3444. MR 0203595 (34:3444)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0290309-X
Article copyright: © Copyright 1972 American Mathematical Society

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