Wang's Liouville conjecture for positive harmonic functions with nonlinear boundary conditions

Let $(M^n,g)$, with $n\geq 3$, be a compact Riemannian manifold with nonnegative Ricci curvature, and let the second fundamental form satisfy $\Pi\geq 1$ on $\partial M$. Let $u\in C^{\infty}(M)$ be positive and satisfy

$$\begin{cases} \Delta u=0 & \text{in }M^n,\\\\ \dfrac{\partial u}{\partial \nu}+\lambda u=u^q & \text{on }\partial M^n, \end{cases}$$

where $1<q\leq \dfrac{n}{n-2}$ and $0<\lambda\leq \dfrac{1}{q-1}$ are constants. Wang's conjecture. Either $u$ is constant, or $q=\dfrac{n}{n-2}$, $\lambda=\dfrac{n-2}{2}$, $(M^n,g)$ is isometric to the unit ball $\mathbb{B}^n\subset\mathbb{R}^n$, and, for some $a\in\mathbb{B}^n$,

$$u(x)=\left(\dfrac{n-2}{2}\dfrac{1-|a|^2}{|a|^2|x|^2-2\langle x,a\rangle+1}\right)^{\frac{n-2}{2}}.$$

This is a Liouville-type rigidity conjecture for harmonic functions with a nonlinear boundary condition. The paper confirms some parameter ranges, while the general case for $n\geq 3$ remains open; the two-dimensional analogue has been resolved.