Some rigidity results related to the Obata type equation
Yiwei Liu, Yihu Yang
TL;DR
This work addresses rigidity phenomena for Obata-type equations on manifolds with boundary under Robin-type data. By exploiting a warped-product structure induced by solutions of $\nabla^2 f - f g = 0$ (and its $+f g$ analogue), and under curvature bounds such as $\operatorname{Ric}^{\Omega} \ge -n$ with $H \ge c$ or suitable boundary-curvature conditions, the authors classify the ambient space as a geodesic ball in $\mathbb{H}^{n+1}$ or as a $Z_2$-symmetric warped-domain over $\Omega_0$, with explicit models $g = dt^2 + (\cosh t)^2 g|_{\Omega_0}$. In the sphere setting, the corresponding rigidity for $\nabla^2 f + f g = 0$ places $\Omega$ as a geodesic ball in $\mathbb{S}^{n+1}$. The approach hinges on the level-set geometry of $f$, the constant $|\nabla f|^2 - f^2$, and the geometry of the boundary, yielding sharp geometric and topological classifications and connecting to eigenvalue-type inequalities via Reilly-type formulas.
Abstract
Let $(Ω^{n+1},g)$ be an $(n + 1)$-dimensional smooth complete connected Riemannian manifold with compact boundary $\partialΩ=Σ$ and $f$ a smooth function on $Ω$ which satisfies the Obata type equation $\nabla^2 f -fg =0$ with Robin boundary condition $f_ν = cf$, where $c=\cothθ>1$. In this paper, we provide some rigidity results based on the warped product structure of $Ω$ determined by the equation $\nabla^2 f -fg =0$ and appropriate curvature assumptions. We also apply a similar method to the Obata type equation $\nabla^2 f +fg =0$ and get a rigidity result on the standard sphere $\mathbb{S}^{n+1}$.