2024/11/27 by Liu, Yiwei, Yang, Yihu
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2411.18508
Let (Ωn+1,g) be an (n + 1)-dimensional smooth complete connected Riemannian manifold with compact boundary ∂Ω=Σ and f a smooth function on Ω which satisfies the Obata type equation ∇2 f -fg =0 with Robin boundary condition fν = 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 ∇2 f -fg =0 and appropriate curvature assumptions. We also apply a similar method to the Obata type equation ∇2 f +fg =0 and get a rigidity result on the standard sphere \mathbbSn+1.