An important problem in differential geometry is to construct conformal metrics on S'2 whose Gauss curvature equals a given positive function f. This problem is equivalent to finding a solution of the equation -Δ_0w = fe~(2w)-1, where Δ_0 denotes the Laplace operator associated with the standard metric g_0 on S~2. J. Moser [22] proved that this equation has a solution if the function f satisfies the condition f(x) = f(-x) for all x ∈ S~2. The general case was studied by S.-Y. A. Chang, M. Gursky, and P. Yang [10, 11, 14].
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机译:微分几何中的一个重要问题是在高斯曲率等于给定正函数f的S'2上构造共形度量。这个问题等同于找到方程-Δ_0w= fe〜(2w)-1的解,其中Δ_0表示与S〜2上的标准度量g_0关联的拉普拉斯算子。 J. Moser [22]证明,对于所有x∈S〜2,如果函数f满足条件f(x)= f(-x),则该方程式具有解。 S.-Y研究了一般情况。 A. Chang,M。Gursky和P. Yang [10,11,14]。
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