Classification of complete gradient conformal mean curvature solitons (Q6597327)

The authors consider a compact \(n\)-dimensional manifold \((M,\partial M,g_{0}) \) with \(n\geq 3\), where \(\partial M\) is the smooth boundary of \(M\) and \(g_{0}\) a Riemannian metric on \(M\). The Yamabe flow with boundary is defined as \[ \frac{\partial }{\partial t}g(t)=-(R_{g}(t)-\overline{R}_{g}(t))g(t)\] on \(M\) and \(H_{g}(t)=0\) on \(\partial M\), with \(g(t)\mid _{t=0}=g_{0}\), \(\overline{R} _{g}(t)\) being the average scalar curvature \(R_{g}(t)\). The conformal mean curvature flow with boundary is defined as \[\frac{\partial }{\partial t} g(t)=-(H_{g}(t)-\overline{H}_{g}(t))g(t)\] on \(M\) and \(R_{g}(t)=0\) on \( \partial M\), with \(g(t)\mid _{t=0}=g_{0}\), \(\overline{H}_{g}(t)\) being the mean curvature \(H_{g}(t)\). \(g(t)\) is a conformal mean curvature soliton if \( g(t)=\sigma (t)\psi _{t}^{\ast }(g0)\) satisfies \(\frac{\partial }{\partial t} g(t)=-H_{g}(t)\) on \(\partial M\) and \(R_{g}(t)=0\) in \(M\), where \(\sigma :[0,T)\rightarrow (0,\infty )\) is a differentiable function such that \( \sigma (0)=1\), and \(\psi _{t}:M\rightarrow M\) is a 1-parameter family of diffeomorphisms in \(M\) such that \(\psi _{0}=id_{M}\). The quadruple \( (M,g_{0},X,\lambda )\) is a conformal mean curvature soliton if \[(\lambda -H_{g_{0}})g_{0}=\mathcal{L}_{X}g_{0}\] on \(\partial M\), \[R_{g_{0}}=0\] on \(M\), and \[\left\langle X,\nu _{g_{0}}=0\right\rangle \] on \(\partial M\), where \( \lambda =-\overset{.}{\sigma }(0)\), \(\mathcal{L}_{X}\) is the Lie derivative with respect to the complete vector field \(X\) generated by \(\psi _{t}\). A conformal mean curvature soliton is called gradient if the vector field \(X\) is a gradient vector field, that is, \[X=\frac{1}{2}\nabla _{g_{0}}f\] for some smooth function \(f\) in \(M\). \N\NThe main result of the paper proves that if \( (M,\partial M,g,f)\) is a nontrivial complete gradient conformal mean curvature soliton, then \((M,\partial M,g,f)\) is a product \((\mathbb{ R},dr^{2})\times (N^{n-1},g)\), where \((N^{n-1},g)\) is scalar flat, with constant mean curvature \(H_{g}\) on the boundary, and \(f=c_{1}r+c_{2}\) for constants \(c_{1},c_{2}\) and \(H_{g}=\lambda c_{1}^{3}\). The authors first prove that every complete nontrivial gradient conformal mean curvature soliton admits a global warped product structure with a \(1\)-dimensional base. They then prove that the scalar curvature \(\overline{R}\) of \((N^{n-1},g)\) satisfies \[ \overline{R}=(n-1)(n-2)(f^{\prime \prime })^{2}+2(n-1)f^{\prime }f^{\prime \prime \prime },\] which implies that \(\overline{R}\) remains constant, and they analyze this equation.