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A problem of prescribing Gaussian curvature on $S^2$

Author(s): Sulbha Goyal; Vinod Goyal
Journal: Proc. Amer. Math. Soc. 129 (2001), 3757-3758.
MSC (2000): Primary 35J30, 35J60; Secondary 31B30
Posted: June 27, 2001
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Abstract:

A class of functions $K(x)=K(x_1,x_2,x_3)$ and the corresponding solutions of

\begin{displaymath}\Delta u + K(x)e^{2u}=1\end{displaymath}

are obtained as a special case of the solutions of

\begin{displaymath}\Delta^mu+K(x)e^{au}=f(x),\qquad x=(x_1,x_2,\dots,x_n),\end{displaymath}

where $\Delta^m$ is defined as $\Delta(\Delta^{m-1})$.

References:

1.
K. Cheng and J. Smoller, Conformal metric with prescribed Gaussian curvature on $S^2$, UAB International Conference on Differential Equations and Mathematical Physics (Abstracts), March 15-21, 1990, p. 48; Trans. Amer. Math. Soc. 336 (1993), 219-251. MR 93e:53044

2.
V. B. Goyal, Remark on a paper of Cheng and Smoller, Proc. Amer. Math. Soc. 113 (1991), 795-797. MR 92b:58243

3.
J. Kazdan and F. Warner, Curvature functions for compact $2$-manifolds, Ann. of Math. (2) 99 (1974), 14-47. MR 49:7949

4.
J. Moser, On a non-linear problem in differential geometry, Dynamical Systems, M. Peixoto (ed.), Academic Press, New York, 1973. MR 49:4018

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Additional Information:

Sulbha Goyal
Affiliation: Department of Mathematics, Tuskegee University, Tuskegee, Alabama 36088

Vinod Goyal
Affiliation: Department of Mathematics, Tuskegee University, Tuskegee, Alabama 36088

DOI: 10.1090/S0002-9939-01-06330-4
PII: S 0002-9939(01)06330-4
Keywords: Laplace operator, Gaussian curvature, conformally equivalent, metric
Received by editor(s): December 20, 2000
Posted: June 27, 2001
Communicated by: David S. Tartakoff
Copyright of article: Copyright 2001, American Mathematical Society

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