Surfaces of Revolution with Monotonic Increasing Curvature and an Application to the Equation $\Delta u = 1 - Ke^{2u}$ on $S^2

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 $\Delta u = 1 - Ke^{2u}$, on $S^2$, has no axially symmetric solutions $u$, given axially symmetric data $K$.

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 1972-03, Vol.32 (1), p.139-141
Main Authors: Kazdan, Jerry L., Warner, Frank W.
Format: Article
Language:English
Subjects:
Axes of rotation
Boundary conditions
Curvature
Embeddings
Mathematical functions
Mathematical monotonicity
Rotation
Surfaces of revolution
Symmetry
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Description
Summary: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 $\Delta u = 1 - Ke^{2u}$, on $S^2$, has no axially symmetric solutions $u$, given axially symmetric data $K$.
ISSN:0002-9939
1088-6826