Immersions of \(r\)-almost Yamabe solitons into Riemannian manifolds (Q2162773)

Let \(g(t)\) be a one-parameter family of Riemannian metrics on a manifold \(M\) and \(S(t)\) its scalar curvature with respect to \(g(t)\). The equation for \textit{Yamabe flow} is given by \(\partial_t g(t) = - S(t)g(t)\). An \(n\)-dimensional (connected) Riemannian manifold \((M, g)\), \(n\ge 2\), is called an \textit{almost Yamabe soliton} if it satisfies \(\mathcal{L}_X g = 2(S - \lambda)g\), where \(\mathcal{L}_X g\) is the Lie derivative of the metric \(g\) with respect to a vector field \(X\) on \(M\), \(\lambda :M\to \mathbb{R}\) is a smooth function and \(S\) denotes the scalar curvature of \(M\). Here, the authors provide conditions under which an almost Yamabe soliton can/cannot be immersed as a codimension-one minimal hypersurface of a Riemannian manifold of either constant or non-positive (strictly negative) sectional curvature.