Let [Formula: see text] be a three-dimensional Lie group with a bi-invariant metric. Consider [Formula: see text] a strictly convex domain in [Formula: see text]. We prove that if [Formula: see text] is a stable CMC free-boundary surface in [Formula: see text] then [Formula: see text] has genus either 0 or 1, and at most three boundary components. This result was proved by Nunes [I. Nunes, On stable constant mean curvature surfaces with free-boundary, Math. Z. 287(1–2) (2017) 73–479] for the case where [Formula: see text] and by R. Souam for the case where [Formula: see text] and [Formula: see text] is a geodesic ball with radius [Formula: see text], excluding the possibility of [Formula: see text] having three boundary components. Besides [Formula: see text] and [Formula: see text], our result also apply to the spaces [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. When [Formula: see text] and [Formula: see text] is a geodesic ball with radius [Formula: see text], we obtain that if [Formula: see text] is stable then [Formula: see text] is a totally umbilical disc. In order to prove those results, we use an extended stability inequality and a modified Hersch type balancing argument to get a better control on the genus and on the number of connected components of the boundary of the surfaces.

中文翻译：

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在双不变李群的严格凸域中的稳定 CMC 自由边界表面上

令 [公式：见正文] 为具有双不变度量的三维李群。考虑 [Formula: see text] 中的 [Formula: see text] 中的严格凸域。我们证明如果[Formula: see text] 是[Formula: see text] 中的稳定CMC 自由边界面，则[Formula: see text] 有0 或1 的属，最多三个边界分量。Nunes [I. Nunes，在具有自由边界的稳定恒定平均曲率表面上，数学。Z. 287(1-2) (2017) 73-479] 用于 [公式：见文本] 的情况，R. Souam 用于 [公式：见文本] 和 [公式：见文本] 是测地线的情况半径为[公式：见文本]的球，排除[公式：见文本]具有三个边界分量的可能性。除了 [Formula: see text] 和 [Formula: see text] 之外，我们的结果也适用于空格 [Formula: see text]，[公式：见正文]、[公式：见正文]和[公式：见正文]。当[公式：见文]和[公式：见文]是一个半径为[公式：见文]的测地线球时，我们得到如果[公式：见文]是稳定的，那么[公式：见文]是一个完全脐光盘。为了证明这些结果，我们使用扩展的稳定性不等式和改进的 Hersch 类型平衡参数来更好地控制属和曲面边界的连通分量的数量。