Is it possible to obtain unbounded minimal surfaces in certain asymptotically flat three-manifolds as a limit of solutions to a natural mountain pass problem with diverging boundaries? In this work, we give evidence that this might be true by analyzing related aspects in the case of the exact Riemannian Schwarzschild manifold. More precisely, we observe that the simplest minimal surface in this space has Morse index one. We prove also a relationship between the length of the boundary and the density at infinity of general minimal surfaces satisfying a free-boundary condition along the horizon.

中文翻译：

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关于黎曼Schwarzschild流形中的自由边界最小曲面

是否有可能在某些渐近平坦的三流形中获得无界的最小曲面，作为对边界发散的自然山pass问题的解决方案的限制？在这项工作中，我们通过分析精确黎曼Schwarzschild流形情况下的相关方面来证明这一点是正确的。更准确地说，我们观察到该空间中最简单的最小曲面的莫尔斯系数为1。我们还证明了边界的长度与满足水平自由边界条件的一般最小曲面的无穷大处的密度之间的关系。