scholarly journals Conformal foliations on Lie groups and complex-valued harmonic morphisms

2021 ◽  
Vol 159 ◽  
pp. 103940
Author(s):  
Elsa Ghandour ◽  
Sigmundur Gudmundsson ◽  
Thomas B. Turner
Keyword(s):  
Lie Groups ◽  
Harmonic Morphisms ◽  
Complex Valued

scholarly journals Holomorphic harmonic morphisms from four-dimensional non-Einstein manifolds

2015 ◽  
Vol 26 (01) ◽  
pp. 1550006 ◽  
Author(s):  
Sigmundur Gudmundsson
Keyword(s):  
Lie Groups ◽  
Hermitian Structure ◽  
Harmonic Morphisms ◽  
Einstein Manifolds ◽  
Open Question ◽  
Complex Valued

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.

Bernstein's theorem for compact, connected Lie groups

1986 ◽  
Vol 99 (2) ◽  
pp. 297-305 ◽  
Author(s):  
Garth I. Gaudry ◽  
Rita Pini
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Image Position ◽  
Complex Valued ◽  
Connected Lie Group

In what follows, G will denote a compact, connected Lie group.Definition. A complex-valued function f ε L2(G) lies in the space A(G) if it can be written in the formwhereThe A(G) norm of f is the infimum of all sums (2) with respect to all decompositions (1).

scholarly journals Proper r-harmonic functions from Riemannian manifolds

2019 ◽  
Vol 57 (1) ◽  
pp. 217-223 ◽  
Author(s):  
Sigmundur Gudmundsson ◽  
Marko Sobak
Keyword(s):  
Lie Groups ◽  
Riemannian Manifolds ◽  
Harmonic Functions ◽  
New Method ◽  
Semisimple Lie Groups ◽  
Complex Valued

AbstractWe introduce a new method for constructing complex-valued r-harmonic functions on Riemannian manifolds. We then apply this for the important semisimple Lie groups $$\mathbf{SO }(n)$$SO(n), $$\mathbf{SU }(n)$$SU(n), $$\mathbf{Sp }(n)$$Sp(n), $$\mathbf{SL }_{n}({\mathbb {R}})$$SLn(R), $$\mathbf{Sp }(n,{\mathbb {R}})$$Sp(n,R), $$\mathbf{SU }(p,q)$$SU(p,q), $$\mathbf{SO }(p,q)$$SO(p,q), $$\mathbf{Sp }(p,q)$$Sp(p,q), $$\mathbf{SO }^*(2n)$$SO∗(2n) and $$\mathbf{SU }^*(2n)$$SU∗(2n).

scholarly journals Harmonic morphisms from four-dimensional Lie groups

2014 ◽  
Vol 83 ◽  
pp. 1-11 ◽  
Author(s):  
Sigmundur Gudmundsson ◽  
Martin Svensson
Keyword(s):  
Lie Groups ◽  
Harmonic Morphisms

scholarly journals Harmonic morphisms from homogeneous spaces of positive curvature

2014 ◽  
Vol 157 (2) ◽  
pp. 321-327
Author(s):  
SIGMUNDUR GUDMUNDSSON ◽  
MARTIN SVENSSON
Keyword(s):  
Homogeneous Space ◽  
Homogeneous Spaces ◽  
Positive Curvature ◽  
Local Existence ◽  
Harmonic Morphisms ◽  
Complex Valued

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).

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