International Journal of Mathematics ( IF 0.604 ) Pub Date : 2020-08-18 , DOI: 10.1142/s0129167x2050086x Ezequiel Barbosa; Farley Santana; Abhitosh Upadhyay
Let be a three-dimensional Lie group with a bi-invariant metric. Consider a strictly convex domain in . We prove that if is a stable CMC free-boundary surface in then 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 and by R. Souam for the case where and is a geodesic ball with radius , excluding the possibility of having three boundary components. Besides and , our result also apply to the spaces , , and . When and is a geodesic ball with radius , we obtain that if is stable then 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.