Conformal solitons to the mean curvature flow and minimal submanifolds (Q2841684)

The authors consider a submanifold \(M\) of a Riemannian manifold \(N\) and the mean curvature flow \((F_t)\): \(dF_t/dt = H_t\), \(H_t\) being the mean curvature vector of \(F_t(M)\) in \(N\). A conformal vector field \(X\) on \(N\) is a conformal soliton whenever its component orthogonal to \(M\) coincides with the mean curvature vector of \(M\) in \(N\). The main results are:NEWLINE{\parindent=2.5cm\begin{itemize}\item[Theorem] A compact self-shrinker in \(\mathbb R^{n+1}\) of codimension \(1\) is stable if and only if it is a sphere. \item[Theorem] The grim reaper is a stable translating soliton in \(\mathbb R^2\).NEWLINENEWLINE\end{itemize}}NEWLINEBesides that, the authors prove the following: {\parindent=2.5cm\begin{itemize}\item[Theorem] If \(X\) is a special conformal soliton and \(N\) is simply connected, then there exists a warped product metric on \(\mathbb R\times N\) such that \(M\subset N\) satisfies the soliton equation if and only if \(\mathbb R\times M\) is minimal in \(\mathbb R\times N\).NEWLINENEWLINE\end{itemize}}