Surfaces of revolution with monotonic increasing curvature and an application to the equation $\Delta u=1-K e^{2u}$ on $S^{2}$
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- by Jerry L. Kazdan and Frank W. Warner PDF
- Proc. Amer. Math. Soc. 32 (1972), 139-141 Request permission
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
- Studies in global geometry and analysis, Mathematical Association of America; distributed by Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0210537
- Jerry L. Kazdan and F. W. Warner, Integrability conditions for $\Delta u=k-Ke^{\alpha u}$ with applications to Riemannian geometry, Bull. Amer. Math. Soc. 77 (1971), 819β823. MR 282314, DOI 10.1090/S0002-9904-1971-12818-5
- Louis Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math. 6 (1953), 337β394. MR 58265, DOI 10.1002/cpa.3160060303
- Barrett OβNeill, Elementary differential geometry, Academic Press, New York-London, 1966. MR 0203595
Additional Information
- © Copyright 1972 American Mathematical Society
- 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