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

    https://doi.org/10.1142/S0129167X2050086XCited by:0 (Source: Crossref)

    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.

    AMSC: 53A10, 53C42, 49Q05, 49Q10, 49Q20

    References

    • 1. L. L. Ahlfors , Open Riemann surfaces and extremal problems on compact subregions, Comment. Math. Helv. 24 (1950) 100–134. CrossrefGoogle Scholar
    • 2. E. Barbosa , On CMC free-boundary stable hypersurfaces in a Euclidean ball, Math. Ann. 372 (2018) 179–187. Crossref, Web of ScienceGoogle Scholar
    • 3. A. Gabard , Sur la representation conforme des surfaces de Riemann a bord et une caracterisation des courbes separantes [Conformal representation of Riemann surfaces with boundary and characterization of the dividing curves], Comment. Math. Helv. 81(4) (2006) 945–964 (in French). CrossrefGoogle Scholar
    • 4. L. Haizhong and X. Changwei , Stability of capillary hypersurfaces in a Euclidean ball, Pacific J. Math. 297(1) (2018) 131–146. Crossref, Web of ScienceGoogle Scholar
    • 5. J. Milnor , Curvatures of left invariant metrics on Lie groups, Adv. Math. 21 (1976) 293–329. Crossref, Web of ScienceGoogle Scholar
    • 6. I. Nunes , On stable constant mean curvature surfaces with free-boundary, Math. Z. 287(1–2) (2017) 473–479. Crossref, Web of ScienceGoogle Scholar
    • 7. J. Ripol , Constant mean curvature hypersurfaces in a Lie group with a bi-invariant metric, Manuscripta Math. 111 (2003) 459–470. Crossref, Web of ScienceGoogle Scholar
    • 8. A. Ros , Stability of minimal and constant mean curvature surfaces with free boundary, Mat. Contemp. 35 (2008) 221–240. Google Scholar
    • 9. A. Ros and E. Vergasta , Stability for hypersurfaces of constant mean curvature with free boundary, Geom. Dedicata 56(1) (1995) 19–33. Crossref, Web of ScienceGoogle Scholar
    • 10. M. Ross , Schwarz P and D surfaces are stable, Differential Geom. Appl. 2 (1992) 179–195. CrossrefGoogle Scholar
    • 11. R. Souam , On stability of stationary hypersurfaces for the partitioning problem for balls in space forms, Math. Z. 224(2) (1997) 195–208. Crossref, Web of ScienceGoogle Scholar
    • 12. B. Totaro , Cheeger manifolds and the classification of biquotients, J. Differential Geom. 61(3) (2002) 397–451. Crossref, Web of ScienceGoogle Scholar
    • 13. G. Wang and C. Xia , Uniqueness of stable capillary hypersurfaces in a ball, Math. Ann. 374 (2019) 1845–1882. Crossref, Web of ScienceGoogle Scholar