International Journal of Mathematics ( IF 0.604 ) Pub Date : 2020-08-18 , DOI: 10.1142/s0129167x2050086x
Ezequiel Barbosa; Farley Santana; Abhitosh Upadhyay

Let $𝔾$ be a three-dimensional Lie group with a bi-invariant metric. Consider $Ω⊂𝔾$ a strictly convex domain in $𝔾$. We prove that if $Σ⊂Ω$ is a stable CMC free-boundary surface in $Ω$ then $Σ$ 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 $𝔾=ℝ3$ and by R. Souam for the case where $𝔾=𝕊3$ and $Ω$ is a geodesic ball with radius $r<π2$, excluding the possibility of $Σ$ having three boundary components. Besides $ℝ3$ and $𝕊3$, our result also apply to the spaces $𝕊1×𝕊1×𝕊1$, $𝕊1×ℝ2$, $𝕊1×𝕊1×ℝ$ and $SO(3)$. When $𝔾=𝕊3$ and $Ω$ is a geodesic ball with radius $r<π2$, we obtain that if $Σ$ is stable then $Σ$ 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.

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