Complex-valued Harmonic Morphisms Research Articles

Conformal foliations on Lie groups and complex-valued harmonic morphisms

We study left-invariant foliations F on Riemannian Lie groups G generated by a subgroup K. We are interested in such foliations which are conformal and with minimal leaves of codimension two. We classify such foliations F when the subgroup K is one of the important SU(2)×SU(2), SU(2)×SL2(R), SU(2)×SO(2) or SL2(R)×SO(2). By this we yield new multi-dimensional families of Lie groups G carrying such foliations in each case. These foliations F produce local complex-valued harmonic morphisms on the corresponding Lie group G.

Harmonic morphisms from 5-dimensional Lie groups

We consider 5-dimensional Lie groups equipped with a left-invariant Riemannian metric. On such groups we construct left-invariant conformal foliations with minimal leaves of codimension 2. These foliations produce complex-valued harmonic morphisms locally defined on the Lie group.

Curvature conditions for complex-valued harmonic morphisms

We study the curvature of manifolds which admit a complex-valued submersive harmonic morphism with either, totally geodesic fibers or that is holomorphic with respect to a complex structure which is compatible with the second fundamental form.We also give a necessary curvature condition for the existence of complex-valued harmonic morphisms with totally geodesic fibers on Einstein manifolds.

Holomorphic harmonic morphisms from four-dimensional non-Einstein manifolds

We construct four-dimensional Riemannian Lie groups carrying left-invariant conformal foliations with minimal leaves of codimension 2. We show that these foliations are holomorphic with respect to an (integrable) Hermitian structure which is not Kähler. We then prove that the Riemannian Lie groups constructed are not Einstein manifolds. This answers an important open question in the theory of complex-valued harmonic morphisms from Riemannian 4-manifolds.

Harmonic morphisms from homogeneous spaces of positive curvature

AbstractWe prove local existence of complex-valued harmonic morphisms from any Riemannian homogeneous space of positive curvature, except the Berger space Sp(2)/SU(2).

Harmonic morphisms from four-dimensional Lie groups

We consider 4-dimensional Lie groups with left-invariant Riemannian metrics. For such groups we classify left-invariant conformal foliations with minimal leaves of codimension 2. These foliations produce local complex-valued harmonic morphisms.

Riemannian Lie groups with no left-invariant complex-valued harmonic morphisms

We study left-invariant complex-valued harmonic morphisms from Riemannian Lie groups. We show that in each dimension greater than 3 there exist Riemannian Lie groups that do not have any such solutions.

Harmonic morphisms from homogeneous Hadamard manifolds

We present a new method for manufacturing complex-valued harmonic morphisms from a wide class of Riemannian Lie groups. This yields new solutions from an important family of homogeneous Hadamard manifolds. We also give a new method for constructing left-invariant foliations on a large class of Lie groups producing harmonic morphisms.

Harmonic morphisms from solvable Lie groups

AbstractIn this paper we introduce two new methods for constructing harmonic morphisms from solvable Lie groups. The first method yields global solutions from any simply connected nilpotent Lie group and from any Riemannian symmetric space of non-compact type and rank r ≥ 3. The second method provides us with global solutions from any Damek–Ricci space and many non-compact Riemannian symmetric spaces. We then give a continuous family of 3-dimensional solvable Lie groups not admitting any complex-valued harmonic morphisms, not even locally.

Harmonic morphisms from the classical non-compact semisimple Lie groups

We construct the first known complex-valued harmonic morphisms from the non-compact Lie groups SL n ( R ) , SU ∗ ( 2 n ) and Sp ( n , R ) equipped with their standard Riemannian metrics. We then introduce the notion of a bi-eigenfamily and employ this to construct the first known solutions on the non-compact Riemannian SO ∗ ( 2 n ) , SO ( p , q ) , SU ( p , q ) and Sp ( p , q ) . Applying a duality principle we then show how to manufacture the first known complex-valued harmonic morphisms from the compact Lie groups SO ( n ) , SU ( n ) and Sp ( n ) equipped with semi-Riemannian metrics.

On the existence of harmonic morphisms from certain symmetric spaces

In this paper we give a positive answer to the open existence problem for complex-valued harmonic morphisms from the non-compact irreducible Riemannian symmetric spaces SL n ( R ) / SO ( n ) , SU ∗ ( 2 n ) / Sp ( n ) and their compact duals SU ( n ) / SO ( n ) and SU ( 2 n ) / Sp ( n ) . Furthermore we prove the existence of globally defined, complex-valued harmonic morphisms from any Riemannian symmetric space of type IV.

Harmonic morphisms from the compact semisimple Lie groups and their non-compact duals

In this paper we prove the local existence of complex-valued harmonic morphisms from any compact semisimple Lie group and their non-compact duals. These include all Riemannian symmetric spaces of types II and IV. We produce a variety of concrete harmonic morphisms from the classical compact simple Lie groups SO ( n ) , SU ( n ) , Sp ( n ) and globally defined solutions on their non-compact duals SO ( n , C ) / SO ( n ) , SL n ( C ) / SU ( n ) and Sp ( n , C ) / Sp ( n ) .

Weierstrass representations for harmonic morphisms on Euclidean spaces and spheres.

We construct large families of harmonic morphisms which are holomorphic with respect to Hermitian structures by finding heierarchies of Weierstrass-type representations. This enables us to find new examples of complex-valued harmonic morphisms from Euclidean spaces and spheres.

HERMITIAN STRUCTURES AND HARMONIC MORPHISMS IN HIGHER DIMENSIONAL EUCLIDEAN SPACES

We construct new complex-valued harmonic morphisms from Euclidean spaces from functions which are holomorphic with respect to Hermitian structures. In particular, we give the first global examples of complex-valued harmonic morphisms from ℝn for each n>4 which do not arise from a Kähler structure; it is known that such examples do not exist for n≤4.