Abstract. By a classical result of Kazdan-Warner, for any smooth signchanging function f with negative mean on the torus (M,gb) there exists a conformal metric g = egb with Gauss curvature Kg = f , which can be obtained from a minimizer u of Dirichlet’s integral in a suitably chosen class of functions. As shown by Galimberti, these minimizers exhibit “bubbling” in a certain limit regime. Here we sharpen Galimberti’s result by showing that all resulting “bubbles” are spherical. Moreover, we prove that analogous “bubbling” occurs in the prescribed curvature flow.

中文翻译：

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圆环上规定曲率流的“冒泡”

摘要。根据 Kazdan-Warner 的经典结果，对于任何在圆环 (M,gb) 上具有负均值的平滑符号变化函数 f，都存在具有高斯曲率 Kg = f 的共形度量 g = egb ，它可以从一个极小值 u Dirichlet's integral in a suitably chosen class of functions. 正如 Galimberti 所示，这些最小化器在某个极限范围内表现出“冒泡”。在这里，我们通过显示所有产生的“气泡”都是球形来锐化 Galimberti 的结果。此外，我们证明了类似的“冒泡”发生在规定的曲率流中。