Wang's uniqueness conjecture for nonlinear boundary problems on convex manifolds

Let $(M^{n},g)$ be a smooth compact Riemannian manifold with boundary $\Sigma$, with $\operatorname{Ric}\geq 0$ and principal curvature tensor $\Pi\geq 1$ on $\Sigma$. Let $u\in C^{\infty}(M)$ be a positive solution of

$$\Delta u=0\quad\text{on }M,\qquad \frac{\partial u}{\partial\nu}+au=u^{q}\quad\text{on }\Sigma.$$

Assume $a>0$ and $1<q\leq n/(n-2)$. Wang's uniqueness conjecture. If $a\leq 1/(q-1)$, then $u$ must be constant unless $q=n/(n-2)$, $M$ is isometric to $\overline{\mathbb{B}^{n}}\subset\mathbb{R}^{n}$, and, for some $\xi\in\mathbb{B}^{n}$, $u$ corresponds to

$$u(x)=\left[\frac{2}{n-2}\frac{1-|\xi|^{2}}{1+|\xi|^{2}|x|^{2}-2x\cdot\xi}\right]^{(n-2)/2}.$$

Here the displayed expression is the exceptional family on the ball.

This extends the ball uniqueness problem to compact manifolds with nonnegative Ricci curvature and boundary principal curvature at least one. The source attributes this formulation to Wang; its resolution is not supplied in the paper excerpt.