Soliton solutions of the mean curvature flow and minimal hypersurfaces (Q2884432)

Let \((M,g)\) be an oriented Riemannian manifold of dimension \(n+1\geq 3\), and \(X\) a vector field in \(M\). An \(X\)-\textit{pseudosoliton hypersurface} is a hypersurface \(f:\Sigma\to M\) such that \(H+X^\bot=0\), where \(H\) is the mean curvature vector of the hypersurface and \(\bot\) denotes the orthogonal projection over the normal bundle of the hypersurface. Equation \(N+X^\bot=0\) is a necessary condition to produce a soliton solution of the mean curvature flow with respect to \(X\) with initial hypersurface \(f:\Sigma\to M\). It is known that solitons with respect to gradient vector fields correspond to minimal hypersurfaces [\textit{G. Huisken}, J. Differ. Geom. 31, No. 1, 285--299 (1990; Zbl 0694.53005)].NEWLINENEWLINENEWLINENEWLINEA Monge-Ampère system in a contact manifold \(N\) of dimension \(2n+1\) with contact one-form \(\theta\) and contact distibution \(D\) is a differential ideal in the exterior algebra of \(N\) generated by \(\theta\), \(d\theta\) and an \(n\)-form \(\varphi\), i.e., the set NEWLINE\[NEWLINE \alpha\wedge\theta+\beta\wedge d\theta+\gamma\wedge\varphi, NEWLINE\]NEWLINE where \(\alpha\), \(\beta\), \(\gamma\) are differential forms in \(N\). Two Monge-Ampère systems on two contact manifolds are equivalent if there is a diffeomorphism identifying the two ideals (hence a contact transformation).NEWLINENEWLINENEWLINENEWLINEIn this paper, the authors associate to the \(X\)-pseudosoliton hypersurface equation on \(M\) a Monge-Ampère system on the unit bundle of \(M\), and they prove in Theorem 2.3 that it is equivalent to the minimal hypersurface Monge-Ampère system on the Riemannian manifold \((M,\bar{g})\) if and only if \(X=\nabla_gu\), in which case \(\bar{g}=e^{-2u}g\).NEWLINENEWLINENEWLINENEWLINEAs the authors remark, Theorem 2.3 is false when \(n=1\).