We employ min-max methods to construct uncountably many, geometrically distinct, properly embedded geodesic lines in any asymptotically conical surface of non-negative scalar curvature, a setting where minimization schemes are doomed to fail. Our construction provides control of the Morse index of the geodesic lines we produce, which will be always less or equal than one (with equality under suitable curvature or genericity assumptions), as well as of their precise asymptotic behaviour. In fact, we can prove that in any such surface for every couple of opposite half-lines there exists an embedded geodesic line whose two ends are asymptotic, in a suitable sense, to those half-lines.

中文翻译：

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渐近圆锥曲面中的最小-最大嵌入测地线

我们使用 min-max 方法在任何非负标量曲率的渐近圆锥曲面中构造无数的、几何上不同的、正确嵌入的测地线，在这种情况下，最小化方案注定要失败。我们的构造提供了对我们生成的测地线的莫尔斯指数的控制，该指数将始终小于或等于 1（在合适的曲率或通用性假设下相等），以及它们精确的渐近行为。事实上，我们可以证明在任何这样的表面上，对于每对相对的半线，都存在一条嵌入的测地线，其两端在适当的意义上与那些半线渐近。