On stable CMC free-boundary surfaces in a strictly convex domain of a bi-invariant Lie group

Barbosa, E and Santana, F and Upadhyay, A (2020) On stable CMC free-boundary surfaces in a strictly convex domain of a bi-invariant Lie group. In: International Journal of Mathematics, 31 (11).

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Abstract

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 < π 2 , 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 < π 2 , 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.

Item Type:	Journal Article
Publication:	International Journal of Mathematics
Publisher:	World Scientific
Additional Information:	The copyright for this article belongs to World Scientific.
Keywords:	Constant mean curvature; Free boundary; Lie group
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	13 Feb 2023 04:36
Last Modified:	13 Feb 2023 04:36
URI:	https://eprints.iisc.ac.in/id/eprint/80202

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