Area-minimizing properties of Pansu spheres in the sub-Riemannian 3-sphere

Ana Hurtado; César Rosales

doi:10.1515/acv-2021-0050

Published online by De Gruyter April 20, 2022

Area-minimizing properties of Pansu spheres in the sub-Riemannian 3-sphere

Ana Hurtado and César Rosales

From the journal Advances in Calculus of Variations

https://doi.org/10.1515/acv-2021-0050

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Abstract

We consider the sub-Riemannian 3-sphere ( 𝕊 3 , g h ) obtained by restriction of the Riemannian metric of constant curvature 1 to the planar distribution orthogonal to the vertical Hopf vector field. It was shown in [A. Hurtado and C. Rosales, Area-stationary surfaces inside the sub-Riemannian three-sphere, Math. Ann. 340 2008, 3, 675–708] that ( 𝕊 3 , g h ) contains a family of spherical surfaces { 𝒮 λ } λ ⩾ 0 with constant mean curvature λ. In this work, we first prove that the two closed half-spheres of 𝒮 0 with boundary C 0 = { 0 } × 𝕊 1 minimize the sub-Riemannian area among compact C 1 surfaces with the same boundary. We also see that the only C 2 solutions to this Plateau problem are vertical translations of such half-spheres. Second, we establish that the closed 3-ball enclosed by a sphere 𝒮 λ with λ > 0 uniquely solves the isoperimetric problem in ( 𝕊 3 , g h ) for C 1 sets inside a vertical solid tube and containing a horizontal section of the tube. The proofs mainly rely on calibration arguments.

Keywords: Sub-Riemannian spheres; Plateau problem; isoperimetric problem; calibrations

MSC 2010: 53C17; 49Q05; 49Q20

Communicated by Zoltan Balogh

Funding source: Ministerio de Economía y Competitividad

Award Identifier / Grant number: MTM2017-84851-C2-1-P

Funding source: Agencia Estatal de Investigación

Award Identifier / Grant number: PID2020-118180GB-I00

Funding source: Junta de Andalucía

Award Identifier / Grant number: A-FQM-441-UGR18

Award Identifier / Grant number: PY20-00164

Funding statement: The authors were partially supported by MEC-Feder grant MTM2017-84851-C2-1-P, the research grant PID2020-118180GB-I00 funded by MCIN/AEI/10.13039/501100011033, and by Junta de Andalucía grants A-FQM-441-UGR18 and PY20-00164.

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Received: 2021-06-09

Revised: 2021-12-09

Accepted: 2022-01-14

Published Online: 2022-04-20

Area-minimizing properties of Pansu spheres in the sub-Riemannian 3-sphere

Abstract

References

Journal