In this short paper, we show a sufficient condition for the solvability of the Dirichlet problem at infinity in Riemannian cones (as defined below). This condition is related to a celebrated result of Milnor that classifies parabolic surfaces. When applied to smooth Riemannian manifolds with a special type of metrics, which generalize the class of metrics with rotational symmetry, we obtain generalizations of classical criteria for the solvability of the Dirichlet problem at infinity. Our proof is short and elementary: it uses separation of variables and comparison arguments for ODEs.

中文翻译：

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对黎曼锥无穷大狄利克雷问题的观察

在这篇简短的论文中，我们展示了狄利克雷问题在黎曼锥体（定义如下）中无穷远的可解性的充分条件。这种情况与 Milnor 对抛物面进行分类的著名结果有关。当应用于具有特殊类型度量的平滑黎曼流形时，它概括了具有旋转对称性的度量类别，我们获得了狄利克雷问题在无穷处可解性的经典标准的概括。我们的证明简短而基本：它使用变量分离和 ODE 的比较参数。