Affiliation:
1. Departamento de Geometría y Topología and Excellence Research Unit “Modeling Nature” (MNat) , Universidad de Granada , Granada , E-18071 , Spain
Abstract
Abstract
We consider the sub-Riemannian 3-sphere
(
𝕊
3
,
g
h
)
{(\mathbb{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
)
{(\mathbb{S}^{3},g_{h})}
contains a family of spherical surfaces
{
𝒮
λ
}
λ
⩾
0
{\{\mathcal{S}_{\lambda}\}_{\lambda\geqslant 0}}
with constant mean curvature λ. In this work, we first prove that the two closed half-spheres of
𝒮
0
{\mathcal{S}_{0}}
with boundary
C
0
=
{
0
}
×
𝕊
1
{C_{0}=\{0\}\times\mathbb{S}^{1}}
minimize the sub-Riemannian area among compact
C
1
{C^{1}}
surfaces with the same boundary. We also see that the only
C
2
{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
{\mathcal{S}_{\lambda}}
with
λ
>
0
{\lambda>0}
uniquely solves the isoperimetric problem in
(
𝕊
3
,
g
h
)
{(\mathbb{S}^{3},g_{h})}
for
C
1
{C^{1}}
sets inside a vertical solid tube and containing a horizontal section of the tube. The proofs mainly rely on calibration arguments.
Funder
Ministerio de Economía y Competitividad
Agencia Estatal de Investigación
Junta de Andalucía
Subject
Applied Mathematics,Analysis
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