Study on (λ,λ)-eigenfunctions on compact manifolds, showing manifold properties and eigenfamily dimensions.
problem Characterizing compact manifolds with (λ,λ)-eigenfunctions and understanding their eigenfamilies. method Analyzing (λ,λ)-eigenfamilies on compact Riemannian manifolds, showing that any such manifold is a mapping torus and any (λ,λ)-eigenfamily is one-dimensional. result Any compact manifold admitting a (λ,λ)-eigenfunction is a mapping torus and any (λ,λ)-eigenfamily is one-dimensional. The study explores globally defined eigenfamilies on closed manifolds, providing existence and orthogonality results.
problem Existence and properties of globally defined eigenfamilies on closed Riemannian manifolds.
method Analyzes topological properties, provides non-/existence results, and uses combinatorial identities.
result Highly rigid orthogonality relations for eigenfunction powers in L2(M). The study characterizes and constructs polynomial harmonic morphisms on spheres.
problem Characterizing and constructing polynomial harmonic morphisms on spheres.
method Characterization and construction of polynomial harmonic morphisms using eigenfamilies.
result Strong restrictions and classification of polynomial harmonic morphisms in low dimensions.
We construct harmonic morphisms on the compact simple Lie group G2. The construction uses eigenfamilies in a representation theoretic scheme.
Constructs explicit p-harmonic functions on Grassmannians and flag manifolds.
problem Finding proper p-harmonic functions on Grassmannians and flag manifolds. method Using the method of eigenfamilies to construct explicit functions.
result Explicit complex-valued proper p-harmonic functions on compact real Grassmannians and non-descending functions on real flag manifolds. New p-harmonic and harmonic morphisms found on Lie groups.
problem Constructing explicit p-harmonic and harmonic morphisms on Lie groups.
method Using the method of eigenfamilies to construct explicit complex-valued p-harmonic functions and harmonic morphisms.
result Explicit complex-valued p-harmonic functions and harmonic morphisms constructed on non-compact classical Lie groups.
Harmonic morphisms and p-harmonic functions constructed on symmetric spaces.
problem Constructing harmonic morphisms and p-harmonic functions on symmetric spaces.
method Using Cartan embedding and related maps to relate tension field and conformality operator.
result Simple formulae relating tension field and conformality operator on symmetric spaces to those on their images.
We construct the first known complex valued harmonic morphisms from the non-compact Lie groups SL(n,R), SU*(2n) 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*(2n), SO(p…