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On stable CMC free-boundary surfaces in a strictly convex domain of a bi-invariant Lie group
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 Ω𝔾 a strictly convex domain in 𝔾. We prove that if ΣΩ 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=3 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 3 and 𝕊3, our result also apply to the spaces 𝕊1×𝕊1×𝕊1, 𝕊1×2, 𝕊1×𝕊1× 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.

更新日期:2020-09-18

 

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