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Keywords:
weighted Riemannian manifold; conformal Killing graph; $f$-mean curvature; Bakry--Émery--Ricci tensor; strong $f$-stability
weighted Riemannian manifold; conformal Killing graph; $f$-mean curvature; Bakry--Émery--Ricci tensor; strong $f$-stability
Summary:
In this paper we investigate the geometry of conformal Killing graphs in a Riemannian manifold $\overline{M}_f^{ n+1}$ endowed with a weight function $f$ and having a closed conformal Killing vector field $V$ with conformal factor $\psi_V$, that is, graphs constructed through the flow generated by $V$ and which are defined over an integral leaf of the foliation $V^{\perp}$ orthogonal to $V$. For such graphs, we establish some rigidity results under appropriate constraints on the $f$-mean curvature. Afterwards, we obtain some stability results for $f$-minimal conformal Killing graphs of $ \overline{M}_f^{ n+1}$ according to the behavior of $ \psi_V$. Finally, related to conformal Killing graphs immersed in $\overline{M}_f^{n+1}$ with constant $f$-mean curvature, we study the strong stability.
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