Bernstein theorems for harmonic morphisms from R3 andS 3

Baird, Paul; Wood, John C.

doi:10.1007/bf01450078

Cited by 69 publications

(125 citation statements)

References 18 publications

Supporting

Mentioning

117

Contrasting

Unclassified

Order By: Relevance

“…We have obtained the following (compare to [3], Theorem 2.15, for the case n = 1) Theorem 3.1. To any holomorphic map from a given Kähler manifold N to the orthogonal Grassmannian and a holomorphic section in the pullback of the universal quotient bundle one can associate a totally geodesic foliation on an Euclidean domain U .…”

Section: Sphh Maps and Sections In Holomorphic Bundlesmentioning

confidence: 83%

“…Foliations whose leaves pass all through the origin correspond to the null-section σ, case (a) in [3], Theorem 2.8, whereas the other foliations correspond to non-trivial sections, case (b) in loc.cit.…”

Section: 2mentioning

confidence: 99%

“…In [3], P. Baird and J. C. Wood have solved completely a problem posed by Jacobi [8]. In modern language, the question was to classify harmonic morphisms with one-dimensional fibres from open domains in R 3 .…”

mentioning

confidence: 99%

See 2 more Smart Citations

Holomorphic vector bundles on Kähler manifolds and totally geodesic foliations on Euclidean open domains

Aprodu

2015

Differential Geometry and its Applications

View full text Add to dashboard Cite

Abstract. In this Note we establish a relation between sections in globally generated holomorphic vector bundles on Kähler manifolds, isotropic with respect to a nondegenerate quadratic form, and totally geodesic foliations on Euclidean open domains. We find a geometric condition for a totally geodesic foliation to originate in a holomorphic vector bundle. For codimension-two foliations, this description recovers [3] Theorem 2.8 and Theorem 2.15. The universal objects that play a key role are the orthogonal Grassmannians.In [3], P. Baird and J. C. Wood have solved completely a problem posed by Jacobi [8]. In modern language, the question was to classify harmonic morphisms with one-dimensional fibres from open domains in R 3 . In other words, it was asked to describe (locally) foliations in lines on open domains in R 3 which verify a geometric extracondition, horizontal weak conformality (HWC). Baird and Wood interpreted horizontal weak conformality as a holomorphic variation of the leaves. More precisely, assuming the foliation was simple, they associate a meromorphic map from the leaf space to a complex projective quadric and indicate an inverse construction. Two distinct cases are found, corresponding to whether or not the leaves pass through the origin.Their approach works for codimension-two foliations on higher-dimensional Euclidean open domains.In this Note, we extend the results of Baird and Wood to foliations of higher codimension. We establish a relation between holomorphic maps to an orthogonal Grassmannian, together with a section in the universal bundle, and certain foliations with totally geodesic leaves on Euclidean open domains, Theorem 3.1, Theorem 3.4. A key feature of the orthogonal Grassmannian is the universal property, it represents the functor of isomorphism classes of isotropic quotient bundles of trivial bundles, Proposition 1.6 in section 1. The definition and the basic properties of the orthogonal Grassmannians are recalled also in section 1.Notation and convention. All the manifolds are considered without boundary. If M is a real manifold, T R M denotes the real tangent bundle, and T C M := T R M ⊗ R C denotes the complexified tangent bundle. For a complex manifold N , we denote by T + N the holomorphic tangent bundle. Recall that, as real vector bundles, T R N and T + N are isomorphic, although they differ as subbundles in T C N . If ϕ : M → N is a submersion Date: February 24, 2018.

show abstract

Section: Sphh Maps and Sections In Holomorphic Bundlesmentioning

confidence: 83%

Section: 2mentioning

confidence: 99%

See 1 more Smart Citation

Holomorphic vector bundles on Kähler manifolds and totally geodesic foliations on Euclidean open domains

Aprodu

2015

Differential Geometry and its Applications

View full text Add to dashboard Cite

show abstract

“…Our main theorem can now be proved by using the Bochner-Yano integral formula (2-7) and the following consequence of [2] and [3]:…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…Our proof of the Main Theorem utilizes a rigidity theorem for shear-free geodesic congruences on S 3 , which follows from the work of P. Baird and J.C. Wood on harmonic morphisms (combining results of [2] and [3]). However, we take the opportunity to provide a direct proof of this rigidity theorem.…”

Section: Introductionmentioning

confidence: 99%

The energy of unit vector fields on the 3-sphere

Higuchi

Kay

Wood

2001

Journal of Geometry and Physics

View full text Add to dashboard Cite

Abstract. The stability of the 3-dimensional Hopf vector field, as a harmonic section of the unit tangent bundle, is viewed from a number of different angles. The spectrum of the vertical Jacobi operator is computed, and compared with that of the Jacobi operator of the identity map on the 3-sphere. The variational behaviour of the 3-dimensional Hopf vector field is compared and contrasted with that of the closely-related Hopf map.Finally, it is shown that the Hopf vector fields are the unique global minima of the energy functional restricted to unit vector fields on the 3-sphere.

show abstract

Harmonic Maps

Chiang¹

2013

Developments of Harmonic Maps, Wave Maps and Yang-Mills Fields Into Biharmonic Maps, Biwave Maps and Bi-Yang-Mills Fields

View full text Add to dashboard Cite

Bernstein theorems for harmonic morphisms from R3 andS 3

Cited by 69 publications

References 18 publications

Holomorphic vector bundles on Kähler manifolds and totally geodesic foliations on Euclidean open domains

Holomorphic vector bundles on Kähler manifolds and totally geodesic foliations on Euclidean open domains

The energy of unit vector fields on the 3-sphere

Harmonic Maps

Contact Info

Product

Resources

About