Poincaré index and the volume functional of unit vector fields on punctured spheres

Brito, Fabiano G. B.; Gomes, André O.; Gonçalves, Icaro

doi:10.1007/s00229-019-01107-y

Cited by 8 publications

(6 citation statements)

References 12 publications

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“…Recently, in 2019, Theorem 3 was extended to odd dimensional spheres S 2n+1 \{±P }, see [3]. In this article, we establish sharp lower bounds for the total area of unit vector fields on antipodally punctured Euclidean sphere S 2 , and these values depend on the indexes of their singularities.…”

Section: Introduction and Main Resultsmentioning

confidence: 83%

Area minimizing unit vector fields on antipodally punctured unit 2-sphere

Brito¹,

Conrado²,

Gonçalves³

et al. 2022

Comptes Rendus. Mathématique

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We provide a lower value for the volume of a unit vector field tangent to an antipodally punctured Euclidean sphere S 2 depending on the length of an ellipse determined by the indexes of its singularities. We also exhibit minimizing vector fields v k within each index class and show that they are the only ones that are sharp for the volume. These fields have areas given essentially by the length of ellipses depending just on the indexes in N and S.

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Section: Introduction and Main Resultsmentioning

confidence: 83%

Area minimizing unit vector fields on antipodally punctured unit 2-sphere

Brito¹,

Conrado²,

Gonçalves³

et al. 2022

Comptes Rendus. Mathématique

Self Cite

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“…The present article is concerned with the geometry of vector fields on the 2-sphere with canonical metric. It brings up some surprising results, in the continuation of [2,3,4,5,6,10] and of course [1].…”

Section: -Previous Resultsmentioning

confidence: 99%

“…A radius r = 1 also brings stability into discussion. This was first and foremost observed in [2,5] in general, hence we assume r = 1 from now on for the meridian type vector fields.…”

Section: -New Vector Fields On the Spherementioning

confidence: 91%

Vector fields with big and small volume on the 2-sphere

Albuquerque¹

2021

Preprint

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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 X m,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, lim k vol(X m,k |Ω ) = +∞. These are the strong candidates to being the minimal volume vector fields in their homology class, since they satisfy great equations. We also show a vector field X on a specific region Ω 1 ⊂ S 2 with volume smaller than any other known optimal vector field restricted to Ω 1 .

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“…The question of minimality in dimension 2 has been studied before and there are several important results eg. in [4][5][6][7]9,14]. A simple differential equation characterizing the 2-dimensional variational problem, ie.…”

Section: Introductionmentioning

confidence: 99%