Proper \(r\)-harmonic functions from Riemannian manifolds (Q2291472)

This article introduces a new method for constructing complex-valued \(r\)-harmonic functions on Riemannian manifolds. The main result is the following. Theorem. Let \(\phi:(M,g)\to \mathbf{C}\) be a complex-valued eigenfunction on a Riemannian manifold such that the tension field \(\tau\) and the conformally operator \(\kappa\) satisfy \[ \tau(\phi)=\lambda \phi\quad\mbox{ and }\quad k(\phi, \phi)=\mu \phi^2, \] for some complex numbers \(\lambda, \mu\). Then, for a natural number \(r\geq 1\) and \((c_1,c_2)\in \mathbb{C}\) any function \[ \Phi_r:W=\{x\in M\mid \phi(x)\not\in (-\infty, 0]\}\to \mathbb{C} \] satisfying \[ \Phi_r(x)= \begin{cases} c_1\log(\phi(x))^{r-1} &\mbox{ if }\mu=0, \lambda\neq 0,\\ c_1\log(\phi(x))^{2r-1}+c_2\log(\phi(x))^{2r-2} &\mbox{ if }\mu\neq 0, \lambda=\mu,\\ c_1\phi(x)^{1-\lambda/\mu}\log(\phi(x))^{r-1}+c_2\log(\phi(x))^{r-} &\mbox{ if }\mu\neq 0, \lambda\neq \mu,\\ \end{cases} \] is proper \(r\)-harmonic on its open domain \(W\) in \(M\). The result is further detailed for standard semisimple Lie groups.