Stability of weighted minimal hypersurfaces under a lower $1$-weighted Ricci curvature bound
Yasuaki Fujitani, Yohei Sakurai
TL;DR
This work develops intrinsic and extrinsic theory for stable $f$-minimal hypersurfaces in weighted manifolds under a lower bound on the $1$-weighted Ricci curvature $ ext{Ric}_f^1$. By deriving a refined stability identity and leveraging comparison geometry, it proves a Schoen--Yau type rigidity criterion, a structural classification for 3D weighted manifolds with $ ext{Ric}_f^1\ge0$, nonexistence results under positive curvature and volume-growth constraints, and a smooth compactness theory for families of $f$-minimal hypersurfaces. The results connect weighted geometry to substatic triples via conformal changes, yielding implications for Lorentzian extrinsic geometry and MOTS stability. Collectively, the paper extends stability and compactness phenomena from the $ ext{Ric}_f^\ ty$ setting to the more delicate $ ext{Ric}_f^1$ regime, with sharp conclusions in dimension three and meaningful structure results in higher dimensions.
Abstract
We will study the $1$-weighted Ricci curvature in view of the extrinsic geometric analysis. We derive several geometric consequences concerning stable weighted minimal hypersurfaces in weighted manifolds under a lower $1$-weighted Ricci curvature bound. We prove a Schoen-Yau type criterion, and conclude a structure theorem for three-dimensional weighted manifolds of non-negative $1$-weighted Ricci curvature. We also show non-existence results under volume growth conditions, and conclude smooth compactness theorems.