• Corpus ID: 239009954

# Vector fields with big and small volume on $\mathbb{S}^2$

@inproceedings{Albuquerque2021VectorFW,
title={Vector fields with big and small volume on \$\mathbb\{S\}^2\$},
author={Rui Albuquerque},
year={2021}
}
We search for minimal volume vector fields on a given Riemann surface, specialising on the case of M?, this is, the 2-sphere with two antipodal points removed. We discuss the homology theory of the unit sphere tangent bundle (SM?, ∂SM?) in relation with calibrations and a minimal volume equation. We find a family Xm,k, k ∈ N, called the meridian type vector fields, defined globally and with unbounded volume on any given open subset Ω of M?. In other words, we have that ∀Ω, limk vol(Xm,k…

## References

SHOWING 1-10 OF 12 REFERENCES
Area-minimizing vector fields on round 2-spheres
• Mathematics
• 2010
Abstract A vector field V on an n-dimensional round sphere Sn (r) defines a submanifold V(Sn ) of the tangent bundle TSn . The Gluck and Ziller question is to find the infimum of the n-dimensional
A critical radius for unit Hopf vector fields on spheres
• Mathematics
• 2006
The volume of a unit vector field V of the sphere (n odd) is the volume of its image V() in the unit tangent bundle. Unit Hopf vector fields, that is, unit vector fields that are tangent to the fibre
Volumes of vector fields on spheres
In this paper we study the problem: What is the unit vector field of smallest volume on an odd-dimensional sphere? We exhibit on each sphere a unit vector field with singularity which has
Poincaré index and the volume functional of unit vector fields on punctured spheres
• Mathematics
manuscripta mathematica
• 2019
For $$n\ge 1$$ n ≥ 1 , we exhibit a lower bound for the volume of a unit vector field on $${\mathbb {S}}^{2n+1}\backslash \{\pm p\}$$ S 2 n + 1 \ { ± p } depending on the absolute values of its
Second variation of volume and energy of vector fields. Stability of Hopf vector fields
• Mathematics
• 2001
Abstract. In this paper we compute the Hessian of the volume of unit vector fields at a minimal one. We also find the Hessians of a family of functionals thus generalizing the known results
Unit vector fields on antipodally punctured spheres: big index, big volume
• Mathematics
• 2008
Nous etablissons une borne inferieure pour le volume d'un champ de vecteurs v defini dans S" \ {±x}, n = 2,3. Cette borne inferieure depend de la somme des valeurs absolues des indices de ? en x et
The volume of a unit vector field in 2 dimensions via calibrations
We use the theory of calibrations to write the equation of a minimal volume vector field on a given Riemann surface.
Algebraic Topology
The focus of this paper is a proof of the Nielsen-Schreier Theorem, stating that every subgroup of a free group is free, using tools from algebraic topology.
Algebraic topology. CUP, Cambridge
• 2002