Almost Yamabe solitons satisfying certain conditions on the associated vector field (Q6606304)

The evolution of a Riemannian metric \(g\) on a smooth manifold \(M\) to a metric \(g(t)\) in time \(t\) through the equation\N\begin{align*}\N\frac{\partial}{\partial t} g(t) &= -r(t)g(t), \\\Ng(0)&=g,\N\end{align*}\Nwhere \(r(t)\) denotes the scalar curvature of \(g(t)\), is called the Yamabe flow. A Yamabe soliton is a special solution of the Yamabe flow that moves by a one-parameter family of diffeomorphisms \(\varphi_{t}\) generated by a time-dependent vector field \(V\) on \(M\) and homotheties, i.e., \(g(t) = \sigma(t)\varphi_{t}^{*}g\) where \(\sigma(t)\) is a positive real valued function of \(t\). This implies that\N\[\N\mathcal{L}_{V}g = 2(r-\rho)g\N\]\Nwhere \(\rho = \frac{1}{2}\sigma'(0)\). The almost Yamabe soliton is said to be shrinking, steady, or expanding according to the sign of \(\rho\). When the constant \(\rho\) becomes a smooth function on \(M\), then a metric satisfying the above equation is called an almost Yamabe soliton.\N\NThe main result of the paper is the following.\N\NTheorem. Let \((M,g,V,\rho)\) be a compact almost Yamabe soliton, with \(n=\mathrm{dim}(M)>2\). Then \(V\) is Killing if any one of the following conditions hold:\N\begin{itemize}\N\item[1.] \(\int_{M} \mathcal{L}_{V}\mathrm{div}(V) dM \geq 0\)\N\N\item[2.] \(V\) is an infinitesimal harmonic transformation, in the sense that \((\mathcal{L}_{V}\nabla)(e_{i},e_{i})=0\), for any local orthonormal frame \(\{e_{1}, \dots, e_{n}\}\) of \((M,g)\).\N\end{itemize}