Area-minimizing properties of Pansu spheres in the sub-Riemannian 3-sphere
Hurtado Cortegana, Ana María
Rosales Lombardo, Manuel César
Sub-Riemannian spheres
Plateau problem
Isoperimetric problem
Calibrations
The authors were supported by MINECO grant MTM2017-84851-C2-1-P and Junta de Andaluc´ıa grants
A-FQM-441-UGR18 and FQM325.
We consider the sub-Riemannian 3- sphere (S-3, gh) 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), no. 3, 675-708] that (S-3, g(h)) contains a family of spherical surfaces {S-lambda}(lambda >= 0) with constant mean curvature.. In this work, we first prove that the two closed half-spheres of S-0 with boundary C-0 = {0} x 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-lambda with lambda > 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.
2022-05-04T11:12:05Z
2022-05-04T11:12:05Z
2021-06-10
info:eu-repo/semantics/article
Published version: Hurtado, A. & Rosales, C. (2022). Area-minimizing properties of Pansu spheres in the sub-Riemannian 3-sphere. Advances in Calculus of Variations, (). [https://doi.org/10.1515/acv-2021-0050]
http://hdl.handle.net/10481/74693
eng
http://creativecommons.org/licenses/by-nc-nd/3.0/es/
info:eu-repo/semantics/openAccess
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