A proper \(p\)-harmonic function on a Riemannian manifold \((M,g)\) is a complex-valued function \(\varphi\colon M\to \mathbb{C}\) satisfying \(\tau^p(\varphi)=0\) and \(\tau^{p-1}(\varphi)\ne 0\), where \(\tau\) is the Laplace-Beltrami operator on \((M,g)\). The main result of the paper is an explicit construction of proper \(p\)-harmonic functions on real Grassmanians \(G_m(\mathbb{R}^{n+m})=\mathrm{SO}(n+m)/\mathrm{SO}(n)\times \mathrm{SO}(m)\).
The construction is based on the following sequence of steps.
(1) If \(\varphi\) is a complex-valued function on \(M\) satisfying \(\tau(\varphi)=\lambda\varphi\) and \(g(\nabla \varphi,\nabla \varphi)=\mu\varphi\) for \(\lambda\ne 0\), \(\mu\ne 0\), then the function \(\Phi_p=(c_1 \varphi^{1-\lambda/\mu}+c_2)\log(\varphi)^{p-1}\) is proper \(p\)-harmonic for any non-vanishing complex coefficients \(c_1,c_2\).
(2) For any \(1\le j,\alpha\le n+m\) the functions \(\hat\varphi_{j\alpha}(X)=\sum_{t=1}^mx_{jt}x_{t\alpha}\) on \(\mathrm{SO}(n+m)\) are \(\mathrm{SO}(n)\times \mathrm{SO}(m)\)-invariant and thus induce functions on the Grassmanian \(G_m(\mathbb{R}^{n+m})\).
(3) For any complex symmetric \((n+m)\times(n+m)\) matrix \(A\), such that \(\operatorname{rank} A=1\), \(\operatorname{tr} A =0\), the \(\mathrm{SO}(n)\times \mathrm{SO}(m)\)-invariant fucntion \(\hat\Phi_A(X) = \sum_{j,\alpha=1}^{n+m} a_{j\alpha}\hat\varphi_{j\alpha}(X)\) on \(\mathrm{SO}(n+m)\) induces a function \(\Phi_A\) on \(G_m(\mathbb{R}^{n+m})\) satisfying condition (1) with \(\lambda=-(n+m)\), \(\mu=-2\).