Some rigidity results related to the Obata type equation

TL;DR

This paper establishes rigidity results for Riemannian manifolds satisfying an Obata type equation with specific boundary conditions, revealing geometric structures and characterizations related to warped products and spheres.

Contribution

It provides new rigidity theorems for manifolds with Obata type equations under boundary conditions, extending understanding of geometric structures and their characterizations.

Findings

Rigidity results for manifolds with $ abla^2 f - fg=0$ under curvature assumptions

Characterization of warped product structures from the Obata equation

Rigidity on the standard sphere for the equation $ abla^2 f + fg=0$

Abstract

Let $(\Omega^{n+1},g)$ be an $(n + 1)$-dimensional smooth complete connected Riemannian manifold with compact boundary $\partial\Omega=\Sigma$ and $f$ a smooth function on $\Omega$ which satisfies the Obata type equation $\nabla^2 f -fg =0$ with Robin boundary condition $f_{\nu} = cf$, where $c=\coth{\theta}>1$. In this paper, we provide some rigidity results based on the warped product structure of $\Omega$ 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 $\mathbb{S}^{n+1}$.