[Paper Review] On f-harmonic morphisms between Riemannian manifolds

Ye‐Lin Ou|arXiv (Cornell University)|Mar 29, 2011

Geometric Analysis and Curvature Flows20 references4 citations

TL;DR

This paper introduces and characterizes $f$-harmonic morphisms—maps between Riemannian manifolds that pull back local harmonic functions to local $f$-harmonic functions. It proves that such maps are precisely horizontally weakly conformal $f$-harmonic maps, generalizing the Fuglede-Ishihara theorem for harmonic morphisms, and provides examples, non-existence results, and connections to conformal geometry and spin systems.

ABSTRACT

f-Harmonic maps were first introduced and studied by Lichnerowicz in \cite{Li} (see also Section 10.20 in Eells-Lemaire's report \cite{EL}). In this paper, we study a subclass of f-harmonic maps called f-harmonic morphisms which pull back local harmonic functions to local f-harmonic functions. We prove that a map between Riemannian manifolds is an f-harmonic morphism if and only if it is a horizontally weakly conformal f-harmonic map. This generalizes the well-known Fuglede-Ishihara characterization for harmonic morphisms. Some properties and many examples as well as some non-existence of f-harmonic morphisms are given. We also study the f-harmonicity of conformal immersions.

Motivation & Objective

• To define and study $f$-harmonic morphisms as a generalization of harmonic morphisms in Riemannian geometry.
• To establish a characterization of $f$-harmonic morphisms in terms of horizontal weak conformality and $f$-harmonicity.
• To explore the existence and non-existence of such maps, particularly on compact and symmetric spaces.
• To connect $f$-harmonic morphisms to physical models, such as inhomogeneous Heisenberg ferromagnets.
• To clarify distinctions between $f$-harmonic morphisms and other related notions like $h$-harmonic morphisms.

Proposed method

• Define $f$-harmonic maps via the critical points of the $f$-energy functional $E_f(\phi) = \frac{1}{2}\int_\Omega f|d\phi|^2 dv_g$.
• Introduce $f$-harmonic morphisms as maps that preserve harmonicity in the $f$-context: pull back local harmonic functions to local $f$-harmonic functions.
• Prove that a map is an $f$-harmonic morphism if and only if it is horizontally weakly conformal and $f$-harmonic, generalizing the Fuglede-Ishihara theorem.
• Use the tension field equation $\tau_f(\phi) = f\tau(\phi) + d\phi(\text{grad}\,f) = 0$ as the defining equation for $f$-harmonic maps.
• Analyze conformal changes of metric: show that $\phi:(M,g)\to(N,h)$ is $f$-harmonic iff $\phi:(M,f^{2/(m-2)}g)\to(N,h)$ is harmonic.
• Apply curvature and submersion theory to prove non-existence results, especially on compact, positively curved manifolds like $S^{2n+1}$.

Experimental results

Research questions

• RQ1 What characterizes $f$-harmonic morphisms among smooth maps between Riemannian manifolds?
• RQ2 How do $f$-harmonic morphisms relate to harmonic morphisms and $p$-harmonic maps?
• RQ3 Under what geometric conditions do $f$-harmonic morphisms exist or fail to exist?
• RQ4 Can $f$-harmonic morphisms exist on compact, positively curved manifolds like $S^{2n+1}$ with non-constant $f$?
• RQ5 What is the physical and geometric significance of $f$-harmonic morphisms, especially in relation to spin systems and conformal geometry?

Key findings

• An $f$-harmonic morphism is characterized as a horizontally weakly conformal $f$-harmonic map, extending the classical Fuglede-Ishihara theorem.
• Any $F$-harmonic map without critical points is an $f$-harmonic map with $f = F'(\frac{|d\phi|^2}{2})$, linking $F$-harmonic and $f$-harmonic maps.
• A polynomial map $\phi:\mathbb{R}^m \to \mathbb{R}^n$ is an $f$-harmonic morphism if and only if it is a harmonic morphism and $f$ has vertical gradient.
• On a compact, non-negatively curved manifold with minimal fibers, a Riemannian submersion is an $f$-harmonic morphism only if $f$ is constant.
• There exists no non-constant positive function $f$ on $S^{2n+1}$ such that the Hopf fibration is an $f$-harmonic morphism.
• For $S^3$ with the standard metric, there is no submersive $f$-harmonic morphism with non-constant $f$ and non-vanishing horizontal curvature, as shown by conformal deformation arguments.