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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A note on the problem of prescribing Gaussian curvature on surfaces


Authors: Wei Yue Ding and Jia Quan Liu
Journal: Trans. Amer. Math. Soc. 347 (1995), 1059-1066
MSC: Primary 53C21; Secondary 35J60, 53A30, 58G30
MathSciNet review: 1257102
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Abstract: The problem of existence of conformal metrics with Gaussian curvature equal to a given function $ K$ on a compact Riemannian $ 2$-manifold $ M$ of negative Euler characteristic is studied. Let $ {K_0}$ be any nonconstant function on $ M$ with $ \max {K_0} = 0$, and let $ {K_\lambda } = {K_0} + \lambda $. It is proved that there exists a $ {\lambda ^{\ast}} > 0$ such that the problem has a solution for $ K = {K_\lambda }$ iff $ \lambda \in ( - \infty ,{\lambda ^{\ast}}]$. Moreover, if $ \lambda \in (0,{\lambda ^{\ast}})$, then the problem has at least $ 2$ solutions.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1995-1257102-2
PII: S 0002-9947(1995)1257102-2
Article copyright: © Copyright 1995 American Mathematical Society