The Obata equation with Robin boundary condition (Q2031496)

For an $n$-dimensional closed Riemannian manifold $(M,g)$ with $\operatorname{Ric}(g)\geq (n-1)g$, \textit{A. Lichnérowicz} [Géométrie des groupes de transformations. Paris: Dunod (1958; Zbl 0096.16001)] proved that the first eigenvalue of the Laplace-Beltrami operator satisfies $\lambda_1\geq n$. Moreover, Obata has shown that the equality $\lambda_1=n$ holds if and only if $(M,g)$ is isometric to the round sphere, as a consequence of the following rigidity result: $(M,g)$ admits a non-constat function $f$ satisfying $$ \nabla^2 g+fg=0 $$ if and only if $(M,g)$ is isometric to the standard sphere. The above displayed equation is known as the Obata equation. We now assume that $M$ has non-empty boundary $\partial M$. The paper under review studies the Obata equation under the Robin boundary condition: \[ \begin{cases} \nabla^2 g+fg=0 & \text{ in }M,\\ \frac{\partial f}{\partial \nu} + af=0&\text{ on }\partial M, \end{cases} \] where $\nu$ is the outward unit normal on $\partial M$ and $a$ is a non-zero constant. As a consequence, the authors obtain that, under certain conditions including $\operatorname{Ric}(g)\geq (n-1)g$, the equality $\lambda_1=n$ holds if and only if $(M,g)$ is a spherical cup (i.e., a geodesic ball in a round sphere).