Research Papers by Sigmundur Gudmundsson
Rendiconti del Circolo Matematico di Palermo Series 2, 2022
Let (G, g) be a 4-dimensional Riemannian Lie group with a 2-dimensional left-invariant, conformal... more Let (G, g) be a 4-dimensional Riemannian Lie group with a 2-dimensional left-invariant, conformal foliation F with minimal leaves. Let J be an almost Hermitian structure on G adapted to the foliation F. We classify such structures J which are almost Kähler (AK) , integrable (I) or Kähler (K). Hereby we construct several new multi-dimensional examples in each class.
Annals of Global Analysis and Geometry, 2023
We construct explicit complex-valued p-harmonic functions and harmonic morphisms on the classical... more We construct explicit complex-valued p-harmonic functions and harmonic morphisms on the classical compact symmetric complex and quaternionic Grassmannians. The ingredients for our construction method are joint eigenfunctions of the classical Laplace-Beltrami and the so-called conformality operator. A known duality principle implies that these p-harmonic functions and harmonic morphisms also induce such solutions on the Riemannian symmetric non-compact dual spaces.
Advances in Geometry, 2023
We use the method of eigenfamilies to construct explicit complex-valued proper p-harmonic functio... more We use the method of eigenfamilies to construct explicit complex-valued proper p-harmonic functions on the compact real Grassmannians. We also find proper p-harmonic functions on the real flag manifolds which do not descend onto any of the real Grassmannians.
Annals of Global Analysis and Geometry, 2024
In this work we construct new multi-dimensional families of compact minimal submanifolds of the c... more In this work we construct new multi-dimensional families of compact minimal submanifolds of the classical Riemannian symmetric spaces SU (n)/SO(n), S p(n)/U(n), SO(2n)/U(n) and SU (2n)/S p(n) of codimension two.
Journal of Geometry and Physics, 2024
Let G be a Lie group equipped with a left-invariant semi-Riemannian metric. Let K be a semisimple... more Let G be a Lie group equipped with a left-invariant semi-Riemannian metric. Let K be a semisimple subgroup of G generating a left-invariant conformal foliation F of codimension two on G. We then show that the foliation F is minimal. This means that locally the leaves of F are fibres of a complex-valued harmonic morphism. In the Riemannian case, we prove that if the metric restricted to K is biinvariant then F is totally geodesic.
Journal of Geometric Analysis, 2024
In this work we introduce a new method for manufacturing minimal submanifolds of Riemannian manif... more In this work we introduce a new method for manufacturing minimal submanifolds of Riemannian manifolds of codimension two. For this we employ the so-called complex-valued eigenfunctions. This is particularly interesting in the cases when the Riemannian ambient manifold is compact. We then give several explicit examples in important cases.
arXiv:2102.07547 , 2021
We apply the method of eigenfamilies to construct new explicit complex-valued p-harmonic function... more We apply the method of eigenfamilies to construct new explicit complex-valued p-harmonic functions on the non-compact classical Lie groups, equipped with their natural semi-Riemannian metrics. We then employ this same approach to manufacture explicit complex-valued harmonic morphisms on these groups.
Rendiconti del Circolo Matematico di Palermo, Series 2, 2023
In this work we construct a variety of new complex-valued proper biharmonic maps and (2, 1)-harmo... more In this work we construct a variety of new complex-valued proper biharmonic maps and (2, 1)-harmonic morphisms on Riemannian manifolds with non-trivial geometry. These are solutions to a non-linear system of partial differential equations depending on the geometric data of the manifolds involved.
Journal of Geometry and Symmetry in Physics, 2022
We study left-invariant foliations F on semi-Riemannian Lie groups G generated by a subgroup K. W... more We study left-invariant foliations F on semi-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), SL2(R), SU(2) × SU(2), SU(2) × SL2(R), SU(2) × SO(2), SL2(R) × SO(2). This way we construct 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.
Journal of Geometry and Analysis, 2022
In this work we construct explicit complex-valued p-harmonic functions on the compact Riemannian ... more In this work we construct explicit complex-valued p-harmonic functions on the compact Riemannian symmetric spaces SU(n)/SO(n), Sp(n)/U(n), SO(2n)/U(n), SU(2n)/Sp(n). We also describe how the same can be manufactured on their non-compact symmetric dual spaces.
Journal of Geometry and Physics, 2021
We construct explicit proper p-harmonic functions on rank-one Lie groups of Iwasawa type. This cl... more We construct explicit proper p-harmonic functions on rank-one Lie groups of Iwasawa type. This class of Lie groups includes many classical Rie-mannian manifolds such as the rank-one symmetric spaces of non-compact type, Damek-Ricci spaces, rank-one Einstein solvmanifolds and Carnot spaces.
Journal of Geometry and Analysis, 2021
We introduce the natural notion of (p, q)-harmonic mor-phisms between Riemannian manifolds. This ... more We introduce the natural notion of (p, q)-harmonic mor-phisms between Riemannian manifolds. This unifies several theories that have been studied during the last decades. We then study the special case when the maps involved are complex-valued. For these we find a characterisation and provide new non-trivial examples in important cases.
Journal of Geometry and Physics, 2021
We study left-invariant foliations F on Riemannian Lie groups G generated by a subgroup K. We are... more We study left-invariant foliations F on Riemannian Lie groups G generated by a subgroup K. We are interested in such fo-liations 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.
Annals of Global Analysis and Geometry, 2020
In this paper we introduce the new notion of complex isoparametric functions on Riemannian manifo... more In this paper we introduce the new notion of complex isoparametric functions on Riemannian manifolds. These are then employed to devise a general method for constructing proper r-harmonic functions. We then apply this to construct the first known explicit proper r-harmonic functions on the Lie group semidirect products R^m*R^n and R^m*H^{2n+1} , where H^{2n+1} denotes the classical (2n+1)-dimensional Heisenberg group. In particular, we construct such examples on all the simply connected irreducible four-dimensional Lie groups.
Journal of Geometry and Physics, 2020
For any positive natural number r ∈ N^+ we construct new explicit proper r-harmonic functions on ... more For any positive natural number r ∈ N^+ we construct new explicit proper r-harmonic functions on the celebrated 3-dimensional Thurston geometries Sol, Nil, SL2(R), H^2 × R and S^2 × R.
Annals of Global Anaysis and Geometry, 2020
We introduce a new method for constructing complex-valued r-harmonic functions on Rie-mannian man... more We introduce a new method for constructing complex-valued r-harmonic functions on Rie-mannian manifolds. We then apply this for the important semisimple Lie groups SO(n), SU(n), Sp(n), SL n (R), Sp(n, R), SU(p, q), SO(p, q), Sp(p, q), SO*(2n) and SU*(2n).
Journal of Geometry and Analysis, 2021
We develop a new scheme for the construction of explicit complex-valued proper bihar-monic functi... more We develop a new scheme for the construction of explicit complex-valued proper bihar-monic functions on Riemannian Lie groups. We exploit this and manufacture many infinite series of uncountable families of new solutions on the special unitary group SU(n). We then show that the special orthogonal group SO(n) and the quaternionic unitary group Sp(n) fall into the scheme. As a by-product we obtain new harmonic morphisms on these groups. All the constructed maps are defined on open and dense subsets of the corresponding spaces.
Journal of Geometry and Physics, 2018
We construct new explicit proper r-harmonic functions on the standard n-dimensional hyperbolic sp... more We construct new explicit proper r-harmonic functions on the standard n-dimensional hyperbolic spaces H^n and spheres S^n for any r ≥ 1 and n ≥ 2.
Journal of Geometry and Physics, 2020
We construct new explicit proper biharmonic functions on the 3-dimensional Thurston geometries So... more We construct new explicit proper biharmonic functions on the 3-dimensional Thurston geometries Sol, Nil, SL2(R), H^2 × R and S^2 × R.
Differential Geometry and its Applications, 2017
We construct new proper biharmonic functions defined on open and dense subsets of the special uni... more We construct new proper biharmonic functions defined on open and dense subsets of the special unitary group SU(2). Then we employ a duality principle to obtain new proper biharmonic functions from the non-compact 3-dimensional hyperbolic space H^3 .
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Research Papers by Sigmundur Gudmundsson