We introduce a new perspective on the classical Nirenberg problem of understanding the possible Gauss curvatures of metrics on $S^{2}$ conformal to the round metric. A key tool is to employ the smooth Cheeger–Gromov compactness theorem to obtain general and essentially sharp a priori estimates for Gauss curvatures $K$ contained in naturally defined stable regions. We prove that in such stable regions, the map $u \to K_{g}$, $g = e^{2u}g_{+1}$ is a proper Fredholm map with well-defined degree on each component. This leads to a number of new existence and non-existence results. We also present a new proof and generalization of the Moser theorem on Gauss curvatures of even conformal metrics on $S^{2}$.

In contrast to previous work, the work here does not use any of the Sobolev-type inequalities of Trudinger–Moser–Aubin–Onofri.

我们介绍了经典 Nirenberg 问题的新视角，即理解 $S^{2}$ 上与圆度量共形的度量可能的高斯曲率。一个关键的工具是使用光滑的 Cheeger-Gromov 紧致定理来获得一般的和本质上尖锐的先验包含在自然定义的稳定区域中的高斯曲率 $K$ 的估计值。我们证明在这样的稳定​​区域中，映射 $u \to K_{g}$, $g = e^{2u}g_{+1}$ 是一个适当的 Fredholm 映射，每个分量都有明确的度数。这导致了许多新的存在和不存在结果。我们还提出了关于 $S^{2}$ 上偶数共形度量的高斯曲率的 Moser 定理的新证明和推广。

与之前的工作相比，这里的工作没有使用 Trudinger-Moser-Aubin-Onofri 的任何 Sobolev 型不等式。